Langmuir 1997, 13, 2227-2234
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Transient Rheology of Wormlike Micelles Jean-Franc¸ ois Berret Unite´ Mixte de Recherche CNRS 5581, Groupe de Dynamique des Phases Condense´ es, Universite´ de Montpellier II, F-34095 Montpellier Cedex 05, France Received November 6, 1996. In Final Form: January 30, 1997X We report on the nonlinear shear rheology of wormlike micelles made of cetylpyridinium chloride (CP+,Cl-) and sodium salicylate (Na+,Sal-) diluted in 0.5 M NaCl-brine. A unique solution at concentration φ ) 12% has been investigated (T ) 20.3 °C). This strongly viscoelastic surfactant solution is an almost perfect Maxwellian fluid in the low-frequency range (ω < 100 rads-1). The stress relaxation function G(t) is decreasing monoexponentially as G(t) ) G0 exp(-t/τR), where G0 ()240 Pa) is the elastic plateau modulus and τR ()1.0 s) the terminal relaxation time. The fundamental feature of the nonlinear rheology is the evidence of a constant and robust stress plateau (no hysteresis) above a characteristic strain rate labeled γ˘ I/N (γ˘ I/N ) 0.9 ( 0.05 s-1). The solution at φ ) 12% was selected because recent flow birefringence experiments revealed that the stress plateau mentioned previously is associated with a nonhomogeneous flow. Two phases of different birefringence and submitted to different velocity gradients have been clearly evidenced in the plateau region. Here we focus on the time dependence of the stress in start-up experiments. Varying the shear rate (γ˘ ) 0.05-10 s-1), we have identified three time ranges corresponding to three kinds of responses of the entangled network of wormlike micelles. At very short time (t , τR), the wormlike micelles reacts as an elastic solid. At times of the order of τR occurs the purely mechanical response of the system. The associated stress as a function of shear rate is interpreted in terms of mechanical instability. Remarkably, this regime exhibits strong similarities with that of conventional polymers: stress overshoot around t ∼ τR and damped oscillations at high strain rates. On the long-time scale (t . τR) and for γ˘ > γ˘ I/N, the system undergoes the transition toward a strongly inhomogeneous flow, which can be ascribed to the isotropic/nematic shear-induced transition. The present findings suggest finally that, in wormlike micellar solution, the purely mechanical instability does exist but is preempted by the transition toward an (nematic) inhomogeneous flow.
1. Introduction There exists by now a challenge, which is to understand the physics of the shear flow properties of wormlike micelles. When subjected to steady shear, surfactant wormlike micelles exhibit a very unusual mechanical response. With increasing shear rates γ˘ , the solution first shear flows as a Newtonian fluid, and then, above a characteristic rate, the shear stress σ(γ˘ ) becomes independent on the shearing field. The flow curve of wormlike micelles, as determined from controlled strain rate rheometry, only shows a true stress plateau, which in some cases can extend over several decades in strain rates. The nature of this behavior as well as the mechanisms at its origin is still the subject of intense debates. Rehage and Hoffmann1 were the first to experimentally point out the existence of the stress plateau in surfactant wormlike micelles. Among the huge variety of materials, a system has become famous through the works of Rehage and Hoffmann.1 It is the ternary system made of cetylpyridinium chloride (CPCl, at a molar concentration of 100 mmol), sodium salicylate (at 60 mmol), and saltfree water. This wormlike micellar system has attracted lots of interest, at least for two reasons. First, the linear low-frequency viscoelasticity is simple: CPCl/Sal (100 mmol/60 mmol) behaves as a Maxwell fluid, with an elastic plateau modulus G0 of 32 Pa and a relaxation time τR of 8 s (η0 ) G0τR ) 256 Pa‚s). The second reason is that, in steady shear experiments, it displays a true stress plateau starting from ∼0.2-0.3 s-1 which persists over almost three decades in strain rates.1 Cates et co-workers2 were the first to recognize the physical importance of the this stress plateau of wormlike micelles on the theoretical side. Very recently, a model X
Abstract published in Advance ACS Abstracts, March 15, 1997.
(1) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712; Mol. Phys. 1991, 74, 933. (2) Spenley, N. A.; Cates, M. E.; MacLeish, T. C. B. Phys. Rev. Lett. 1993, 71, 939.
S0743-7463(96)01078-5 CCC: $14.00
Figure 1. Schematical representation of a nonmonotonic constitutive relation linking the shear stress and the rate of strain γ˘ . In the range [γ˘ M, γ˘ m] (dotted part of the curve), the flow is unstable.
has been proposed on the basis of a nonmonotonic constitutive equation relating σ and γ˘ . Nonmonotonic means here that there exists a γ˘ -range over which the stress is multivalued. A schematic representation is shown in Figure 1. If the applied shear rate lies in the region of decreasing stress (dotted part of the curve in Figure 1, γ˘ M < γ˘ < γ˘ m), an initially homogeneous flow becomes mechanically unstable. As a result, the solution evolves up to a stationary state of shearing where bands of highly sheared liquid of low viscosity coexist with a more viscous part supporting a lower rate. In the banded regime, changes in shear rates essentially alter the proportions of the low- and high-viscosity bands. The flow curve in Figure 1 has been constructed from a microscopic model of wormlike micelles including the reversible breaking of the unidimensional objects, the reptation dynamics, and a solvent contribution (∝γ˘ ), which is essential for the stress upturn at high shear rate (γ˘ > γ˘ m). © 1997 American Chemical Society
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Figure 2. Schematical flow curves for wormlike micelles as obtained from mechanical-rate controlled experiments.3-5,8,9 The surfactant concentration is assumed to be close but below the isotropic-to-nematic phase boundary (in the absence of flow). The low- and high-shear-rate branches (regions I and III) are characterized by a truly homogeneous flow, isotropic and nematic, respectively. In the plateau region (II), the flow is nonhomogeneous (biphasic) and the rate acts on the relative proportions of each phase, isotropic and nematic.
It thus describes a purely mechanical instability and consequently will be referred to in the following. An alternative explanation has been proposed to account for the stress plateau featured in the mechanical response of wormlike micelles. This approach was based on experimental investigations of concentrated solutions close to an isotropic/nematic phase boundary (in the absence of flow). Using small angle neutron scattering3-5 as well as birefringence under shear,5-6 the plateau was interpreted in these cases as the coexistence line of two thermodynamically stable phases present within the sheared solution. These two phases are isotropic and nematic and coexist all along the plateau. It should be pointed out that some rheological data exhibit a power law instead of a true plateau in the two-phase region, the shear stress varying as σ ∼ γ˘ a with exponents a ) 0.1-0.2.3-5 A schematic representation of the nonlinear mechanical response of concentrated wormlike micelles is shown in Figure 2. The low- and high-shear-rate branches (domains I and III in Figure 2) are characterized by a truly homogeneous flow, isotropic and nematic, respectively. In between, in the plateau region (II), the flow is nonhomogeneous (biphasic) and the rate acts on the relative proportions of each phase. Therefore, we are here concerned with a phase transition of first-order induced by shear in the definition of the thermodynamics. On application of a shearing field, an isotropic disordered fluid undergoes a transition toward a nematic oriented state (nonconserved order parameter). Mechanisms of the transition in concentrated solutions of wormlike micelles are poorly understood on the theoretical side.7 The aim of the present paper is to reconcil both approaches evoked above. The stress plateau in the rheological response of wormlike micelles is actually due to a nonequilibrium phase transition triggered by shear, (3) Schmitt, V.; Lequeux, F.; Pousse, A.; Roux, D. Langmuir 1994, 10, 955. Schmitt, V. The`se de Doctorat, Universite´ de Strasbourg, 1994, unpublished. (4) Berret, J.-F.; Roux, D. C.; Porte, G.; Lindner, P. Europhys. Lett. 1994, 25, 521. Roux, D. C. The`se de Doctorat, Universite´ de Montpellier, 1995, unpublished. (5) Cappelaere, E.; Berret, J.-F.; Decruppe, J.-P.; Cressely, R.; Lindner, P. To be published. (6) Cappelaere, E.; Cressely, R.; Decruppe, J.-P. Colloids Surf. 1995, 104, 353-374. Cappelaere, E. The`se de Doctorat, Universite´ de Metz, 1995, unpublished. (7) Schmitt, V.; Marques, C.; Lequeux, F. Phys. Rev. E 1995, 52, 4009.
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but this thermodynamic transition and the purely mechanical instability are intertwined phenomena. In this paper, we report on the shear rheology of wormlike micelles made of cetylpyridinium chloride (CPCl) and sodium salicylate (NaSal) diluted in 0.5 M NaCl-brine. A unique solution at concentration φ ) 12% has been investigated (T ) 20.3 °C). This strongly viscoelastic surfactant solution (G0 ) 240 Pa, τR ) 1 s) has been subjected to steady shear in the range γ˘ ) 0.05-10 s-1. This system was selected for several reasons: One is that, provided that enough time is left to the system to reach its stationary state, the long-time shear stress exhibits a true plateau (γ˘ > γ˘ I/N ) 0.9 ( 0.05 s-1).8,9 A second reason is that flow birefringence experiments performed recently on the same solution (CPCl/Sal, φ ) 12%9) reveal that the plateau is associated with a nonhomogeneous flow. Two phases of different birefringence and submitted to different velocity gradients appear at the onset of the plateau region. So we know a priori for that system that the high-shear properties are dominated by a nonhomogeneous flow. We here focus on the time dependence of the stress in start-up experiments. Varying the shear rate, we have identified three time ranges corresponding to three kinds of shear responses. At very short time (t , τR), the wormlike micelles react as an elastic solid (σ ∼ γ). At times of the order of τR occurs the purely mechanical response of the system. The associated stress as a function of shear rate is interpreted in terms of mechanical instability. Finally, at infinite time and for γ˘ > γ˘ I/N, the system has undergone the transition toward a strongly inhomogeneous flow, which can be ascribed to the isotropic/ nematic shear-induced transition. 2. Experimental Section The surfactant system investigated here is the ternary solution made of cetylpyridinium chloride (CP+,Cl-) and sodium salicylate (Na+,Sal-) (hereafter abbreviated as CPCl/Sal) diluted in 0.5 M NaCl-brine. This system is known to easily form elongated wormlike micelles.1,10 Following our first reports on the CPCl/ Sal system,8,9 we focus here on the transient mechanical responses of a unique solution at total surfactant concentration φ ) 12% and ambient temperature (T ) 20.3 °C). For the present study, the samples were prepared at the constant molar ratio R ) [Sal]/ [CPCl] ) 0.5. The phase diagram of CPCl/Sal solutions has been discussed in detail in ref 8 and will not be documented further. It is however essential to remember that the 12% solution placed under scrutiny here falls in the intermediate concentration regime. It is slightly above the semidilute concentration regime (φ* ) 0.3% < φ < 6%, φ* being the overlap concentration), but it is also far from the isotropic/nematic phase boundary (φI/N ∼ 36%). The linear and nonlinear viscoelastic properties of the CPCl/ Sal solution were obtained on a Rheometrics fluid spectrometer (RFS II) working in a cone-and-plate configuration with controlled shear rate (diameter, 50 and 30 mm; angle, 0.02 rad). Dynamical measurements were carried out for angular frequencies ω ) 0.1100 rad‚s-1 with typical deformation amplitudes of 10%. Steady shear measurements were performed in the range γ˘ ) 0.05-10 s-1. All the data displayed here were obtained according to startup procedures. At t ) 0, shear is applied to the sample initially at rest and the stress response is recorded over time up to typically 1000 s. As long as enough time was let to the sample relax between two consecutive measurements, no memory or hysteresis effects could be detected.
3. Experimental Results As already emphasized in a recent report on the rheology of CPCl/Sal wormlike micelles,9 the solution that we (8) Berret, J.-F.; Roux, D. C.; Porte, G. J. Phys. II 1994, 4, 1261. (9) Berret, J.-F.; Porte, G.; Decruppe, J.-P. Phys. Rev. E, to appear. (10) Hoffmann, H.; Platz, G.; Rehage, H.; Schorr, W.; Ulbricht, W. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, 255. Hoffmann, H.; Kalus, J.; Thurn, H.; Ibel, K. Ber. Bunsenges. Phys. Chem. 1983, 87, 1120.
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Figure 3. Transient shear stress as obtained from the startup experiment (γ˘ ) 0.1 s-1) on the wormlike micellar solution made of cetylpyridinium chloride (CP+,Cl-) and sodium salicylate (Na+,Sal-) in 0.5 M NaCl-brine. The surfactant concentration is φ ) 12%, and the temperature is T ) 20.3 °C. The continuous line is from eq 2 using η0 ) 243 Pa‚s and τR ) 1.3 s.
investigate is an almost perfect Maxwell fluid in the lowfrequency range (ω < 100 rad‚s-1). The stress relaxation function G(t) is decreasing monoexponentially as G(t) ) G0 exp(-t/τR) where G0 is the elastic plateau modulus and τR is the terminal relaxation time. At T ) 20.3 °C, one gets G0 ) 240 ( 5 Pa and τR ) 1.02 ( 0.05 s, yielding a static viscosity η0 ) G0τR ) 240 ( 20 Pa‚s. In the following, the value of τR is considered as the reference time for the transient rheological responses. The terminology “shorttime scale” refers to times of the order of τR or a few τR (typically t < 10 s), whereas on the other hand “long-time scale” will be applied to minutes or more. Figure 3 shows the short-time transient response of the φ ) 12% solution submitted to a steady shear at γ˘ ) 0.1 s-1. At t , τR, the stress increases linearly and then saturates for t . τR at the constant value of 24 Pa. At higher strain rates, short- and long-time transient data are considered separately, as is illustrated in Figure 4 for γ˘ ) 0.8 s-1. Figure 4a exhibits the shear stress up to 12 s after inception of the flow, and Figure 4b exhibits that up to 100 s. The start-up σ(t) increases again linearly with time, passes through a maximum (overshoot) at t ) 2.7 s, and then decreases to a stable value around 130 Pa. Figure 4b emphasizes on the other hand that, for times larger than 10 s, the stress remains constant. At 1.2 s-1 (Figure 5a), the overshoot is still present and even reinforced compared to the 0.8 s-1 data. The main difference with the preceding rate occurs on the long-time range (Figure 5b). After σ(t) stabilizes at a constant level (155.6 Pa) around t ∼ 10 s, the stress decreases now slowly over the next 300 s and reaches a time-independent value, noted σI/N in refs 4 and 8. For the present wormlike solution, σI/N ) 140 Pa. The transient response at γ˘ ) 1.2 s-1 reveals two relaxations which are well separated in time and both characterized by a decrease of the shear stress. One occurs on the τR time scale (overshoot), and the second one takes several minutes to be completed. This latter relaxation will be referred to in the following as the sigmoidal kinetics.11 The time response at 2 s-1 reveals new features. In Figure 6a, where the short-time data are plotted, the stress still exhibits a huge overshoot (11) This terminology of sigmoidal kinetics is borrowed from the field of the dynamics of phase transitions of first order. More precisely, it finds its origin in the time evolution of the proportion of the nucleating and growing phase, which looks like an S-function (sigmoid).
Figure 4. (a, top) Same as in Figure 3 for γ˘ ) 0.8 s-1. The continuous line is from the linear predictions (eq 2). The deviation occurring above t ∼ τR indicates the nonlinear shearthinning behavior. (b, bottom) Time evolution of the transient shear stress at γ˘ ) 0.8 s-1. The long-time scale here emphasizes the overshoot shortly after inception of the flow and the constancy of the stress up to 100 s.
at t ) 1.6 s (it culminates up to 300 Pa), but it is followed now by an undershoot. A closer inspection of Figure 6a reveals indeed σ(t) oscillations. These oscillations are also visible in Figure 6b, where they are followed by a longtime sigmoidal relaxation of the stress toward the steadystate plateau value σI/N ) 140 Pa. Note that this longtime evolution takes now ∼40 s, while in Figure 5b the stationary stress limit was reached within ∼300 s. At 5 s-1, the stress response is almost entirely dominated by oscillations (Figure 7). The first overshoot is again present and arrives at stresses larger than 1500 Pa. This enormous first maximum is followed by several damped oscillations, ending up finally with a steady-state stress value at σI/N ) 140 Pa. The long-time sigmoidal kinetics evidenced previously for γ˘ ) 1.2 and 2 s-1 has now shifted to the short-time scale and as a result is not experimentally to be distinguished from the oscillations. As Figures 3-7 have shown, the transient rheology of CPCl/Sal wormlike micelles is rather complex. In order to provide a readable picture of these transients, at best on a single plot, we have selected three particular instants in the time response. To each of these times is associated a value of the shear stress, as defined below. (i) tos denotes the time of the first overshoot, and σos, the related stress. At all rates, tos falls in the τR range.
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Figure 5. Transient shear stress displayed as in Figure 4, for γ˘ ) 1.2 s-1. (a, top) On the short-time scale. The stress overshoot is now reinforced with respect to the 0.8 s-1 data. (b, bottom) On the long-time scale. The stress exhibits a slow sigmoidal relaxation toward the stationary stress value σI/N ) 140 Pa. The arrows indicate three remarkable values of the shear stress discussed in the text, σOS, σM, and σST.
(ii) σM is the value of the stress after the first overshoot and before the onset of the long-time sigmoidal relaxation. In most cases (low γ˘ ), it corresponds to a typical time noted tM ∼ 10 s. At high strain rates, the sigmoidal relaxations fall in the 10 s range, and then, σM is obtained from the extrapolation of σ(t) as t goes to 0. For reasons which will appear later, σM is defined as the mechanical stress. In analogy with the schematical nonmonotonic behavior of Figure 1, γ˘ M still defines the strain rate at which σM(γ˘ ) passes through a maximum. (iii) The third time of interest is the stationary time tST. It corresponds to the stationary state of shear determined by a time-independent stress, σ(t) ) σST. As an example, the three fundamental values of the stress σos, σM, and σST pointed out above are identified in Figure 5b by arrows. Note of course that not all times and stresses are available for each shear rate. Figure 8 displays the overall flow behavior of the CPCl/ Sal wormlike micelles in terms of σos(γ˘ ), σM(γ˘ ), and σST(γ˘ ). At low shear rates, no overshoot is detected and the mechanical and stationary stresses are equal. Only for rates above ∼1 s-1 do the different stresses as defined above begin to deviate markedly from each other. The overshoot stress σos exhibits a strong increase with rate in the semilogarithmic representation of Figure 8, while σST remains strictly constant at the value already specified,
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Figure 6. Transient shear stress displayed as in Figure 4, for γ˘ ) 2 s-1. (a, top) On the short-time scale. The stress overshoot transforms into oscillations. (b, bottom) On the long-time scale. The sigmoidal relaxation toward the stationary state of flow merges into the oscillations. As compared to the previous responses, the stationary value of the stress remains identical at σI/N ) 140 Pa (indicated by the straight line).
Figure 7. Transient shear stress as a function of time as observed at γ˘ ) 5 s-1. After inception of the flow, the stress exhibits an enormous overshoot culminating around 1500 Pa (not shown in the figure) followed by clearly discernible oscillations (period T ) 4 s). The straight line points out the long-time stationary stress at 140 Pa.
σI/N ) 140 Pa. This latter result is in full agreement with our previous reports.8,9 Above a characteristic rate noted
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is reached if the minimum deformation needed for the sample to reach the stationary state (i.e. σ(t) ) σST) is less than or equal to γlin. For a Maxwell fluid, considering that the stress relaxation function G(t) is less than 1% of its initial starting value G0 for t ∼ 5τR, the above statement is equivalent to
γ(τ∼5τR) ) 5γ˘ τR e γlin
(1)
Using for the present wormlike micelles γlin ) 50% and τR ) 1 s, eq 1 holds for γ˘ e 0.1 s-1, and in that case only, the shear stress follows an exponential increase in startup experiment, according to
σ(t) ) η0γ˘ (1 - exp(-t/τR))
Figure 8. Strain rate variations of three remarkable values of the shear stress observed in the transient responses of the CPCl/Sal wormlike micelles and defined in the text as σOS, σM, and σST. σOS is the stress at the first overshoot (t ) tos), σM characterizes the stress within the material before the onset of the long-time sigmoidal relaxation, and σST corresponds to the stationary state of shear. Note that along the stress plateau one has σST(γ˘ ) ) σI/N ) 140 Pa. Also included for comparison we display the Newtonian behavior calculated using the static viscosity η0 ) 240 Pa‚s (see dotted line). The continuous lines through the data points are guides for the eyes only.
γ˘ I/N, estimated here at 0.9 ( 0.05 s-1, a σ plateau is obtained provided that enough time is left for the system to reach its stationary state. Also included for comparison in Figure 8 we display the Newtonian behavior calculated using the static viscosity η0 ) 240 Pa‚s (see dotted line). It compares approximately with the strain rate variation of the overshoot stress σos(γ˘ ). More interestingly, the mechanical stress data σM(γ˘ ) which first match the stationary values below γ˘ I/N, are found to pass through a maximum at γ˘ M ) 2 ( 0.2 s-1. This maximum coincides with the onset of oscillations at short times. Due to the extremely large amplitudes of these oscillations (Figures 6 and 7), σM could be determined accurately up to γ˘ ) 3 s-1. 4. Analysis and Discussion The transient rheological behaviors of the CPCl/Sal solution as shown in Figures 3-7 are now analyzed as a function of the shear rate. Four γ˘ ranges are determined and discussed with respect to two fundamental parameters, the inverse Maxwell relaxation time 1/τR and the transition rate γ˘ I/N. We will first discuss the linear viscoelastic regime valid for γ˘ τR , 1 and then the nonlinear regime of flow for which the product γ˘ τR is of the order of unity, with three subsections: (i) γ˘ < γ˘ I/N, (ii) γ˘ I/N < γ˘ e γ˘ M, and (iii) γ˘ > γ˘ M. 4.1. Linear Response: γ3 τR , 1. The linear response of a viscoelastic fluid can be achieved in steady shear experiments, under the condition that the total deformation imposed on the solution remains small.12 Three quantities have to be compared in this case: the shear rate, the terminal relaxation time τR (i.e. the Maxwell time), and the limiting deformation noted γlin. γlin is usually determined from dynamical experiments and specifies the range of linear viscoelasticity. For the present CPCl/Sal system, γlin was estimated to be 50 ( 10%. One can show practically that the linear regime in steady shear (12) Larson, R. G. Constitutive Equations for Polymer Melts and Solutions; Butterworths: Boston, 1988.
(2)
Equation 2 has been plotted in Figure 3 in order to compare with the γ˘ ) 0.1s-1 data. The continuous line in Figure 3 is obtained using η0 ) 243 P‚as and τR ) 1.3 s, in fair agreement with the dynamical mode results. 4.2. Nonlinear Response: γ3 τR ∼ 1 and γ3 < γ3 I/N. So, above typically 0.1 s-1, the shear properties exhibit nonlinear effects. The first obvious evidence of the nonlinearity of the stress response is that eq 2 is no longer obeyed. The linear regime prediction is shown for comparison in Figure 4a, where agreement between σ(t,γ˘ )0.8 s-1) and eq 2 is found only at short times, t < τR. Deviations from the linear behavior occur above and result in a shear thinning effect. As in concentrated polymer solutions and melts, the appearance of a stress overshoot on the short-time scale is closely related to the shear thinning of the material. Documented here in Figures 4-7, this overshoot persists at all shear rates investigated, starting at γ˘ ) 0.5 s-1 and going up to γ˘ ) 10 s-1. In the intermediate range, that is between the limit of validity of eq 2 and ∼0.5 s-1, the viscoelasticity is nonlinear but no overshoot is detected. Although less extensively investigated than in polymers,13 stress overshoots have been observed too in surfactant solutions of elongated micelles submitted to start-up experiments, e.g. in CTAB-NaSal/W (cetyltrimethylammonium bromide with sodium salicylate diluted in water).14 The features obtained by Shikata et al. on this latter system compare very well with the present data. Figure 9 displays the shear rate dependence of the deformation γos ) γ˘ tos associated to the stress overshoot. At the onset of the nonlinear regime, for moderate strain rates, γos starts at the constant value of ∼2 and then increases continuously with increasing γ˘ . Figure 9 deserves several comments. First, γos(γ˘ ) is not sensitive to the transition observed on the stationary values of the stress, when, at γ˘ I/N, σST(γ˘ ) becomes shear rate independent. Second, it is interesting to notice here that the γoslimiting value at the onset of nonlinearity is the same as the one measured for entangled solutions of linear and even branched polymers by Menezes and Graessley.13 Actually, the overall variation of the overshoot strain as depicted in Figure 9 resembles strongly that obtained for unbreakable polymers. More strikingly, this limiting value γos ∼ 2 is also the one predicted by the Doi-Edwards constitutive equation based on reptation. This agreement might be fortuitous, since the micellar dynamics involves reptation as well as reversible breaking of the polymer-like chains.15 The above findings suggest however that the constitutive (13) Menezes, E. V.; Graessley, W. W. J. Polym. Sci. 1982, 20, 1817. (14) Shikata, T.; Hirata, H.; Takatori, E.; Osaki, K. J. Non-Newtonian Fluid Mech. 1988, 28, 171. Shikata, T. Ph.D. thesis of the University of Osaka, Japan, 1988, unpublished. (15) Cates, M. E. Macromolecules 1987, 20, 2289.
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Figure 9. Strain rate dependence of the deformation γOS ) γ˘ tOS associated to the first stress overshoot for CPCl/Sal, φ ) 12%. At the onset of nonlinear regime (shown by the arrow, see also the text), the deformation at which the stress overshoot occurs starts at the constant value of γOS ∼ 2. This value of 2 has been found for entangled solutions of linear and branched polymers13 and is also predicted from the reptation dynamics.16
equation describing the nonlinear mechanical behavior of the entangled mesh of wormlike micelles should bear strong similarities with the equivalent equation for polymer. For instance, in the case of the separable timestrain constitutive equation of K-BKZ form, the damping function h(γ) for micelles should be such as the product {γh(γ)} goes through a maximum for strain γ ∼ 2.16 4.3. Plateau Regime: γ3 τR ∼ 1 and γ3 I/N < γ3 e γ3 M. We here focus on the long-time relaxation (long compared to τR) illustrated in Figures 5b and 6b. As already described, this long-time kinetics starts very precisely at γ˘ I/N ) 0.9 ( 0.05 s-1 and can be followed up to ∼3 s-1, the strain rate where damped oscillations definitively dominate the stress response. The transient sigmoidal decrease of the stress in the plateau regime has been already identified, discussed, and analyzed in our preceding reports.8,9,17 This is a feature of the CPCl/Sal wormlike micelles which is found over a broad range of concentration (6% < φ < 31%) and which is always connected to a discontinuity of slope of the stress behavior (see Figure 2).8 It should be emphasized that, in contrast to the short-time response discussed previously (overshoot), it is not a feature shared in common with polymer solutions. Long-time transients have been also reported in other micellar systems.6,17 We are reminded here of the basic interpretation of the long-time sigmoidal kinetics in wormlike micelles. Very recently, we have recognized8 that the stress plateau could be interpreted in terms of a coexistence line of two different thermodynamical phases coexisting in the sheared solution. These two phases were originally speculated to be isotropic (entangled network of micelles) and nematic (highly oriented), and thus the discontinuity in the shear stress behavior was marking the first-order isotropic-to-nematic phase transition induced by shear. It is only very recently that these assumptions could be confirmed experimentally for the CPCl/Sal system.9 To do so, we have examined the state of shearing at the stress plateau in flow birefringence. A rheo-optical device was used which enables the visualization between crossed (16) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986. (17) Berret, J.-F.; Roux, D. C.; Porte, G. Proceedings of the IVth European Conference on Rheology, Seville, 1994, Gallegos, C., Ed.; pp 582.
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polarizers of the gap of a Couette cell, and the solution placed under scrutiny was precisely CPCl/Sal at φ ) 12%. As a result, we have found out that in the plateau regime two separated macroscopic regions of very different birefringence coexist: a very bright birefringent band is located close to the inner cylinder whereas the remaining part of the sample appears to be dark between the polarizers. In the birefringent bright band, the wormlike micelles are strongly oriented with respect to the flow. This region supports a much larger shear gradient than the rest of the sample, and its relative proportion (which increases with γ˘ ) is determined to accommodate the imposed macroscopic velocity gradient. From this study,9 it was concluded that the stress plateau does correspond to an inhomogeneous shear flow of the wormlike micelles where phases differing in viscosity, orientation, and order parameters coexist. The assignment of the oriented band to a shear-induced nematic statesas originally suggesteds will be entirely convincing after closely comparing the data received from flow birefringence and small-angle neutron scattering (measurement of orientational order parameter). Another strong argument in favor of the transition toward an inhomogeneous flow above γ˘ I/N was the formal analogy of the transient rheology obtained on two surfactant wormlike micelles, the present CPCl/Sal systems and the concentrated CPCl/hexanol/brine solutions at φ ) 32%. For this latter system, the I/N transition was clearly established.4 Transient stresses recorded under start-up conditions at shear rates above γ˘ I/N were shown to be analytically similar. The time evolution of the shear stress could be fitted according to the stretched exponential8
[ ( )]
σ(t,γ˘ ) ) σST + ∆σ0 exp -
t τNG(γ˘ )
R
(3)
where ∆σ0 ) σM - σST, τNG(γ˘ ) denotes the characteristic time of the sigmoidal decrease, and R ) 2. In eq 3, the initial amplitude ∆σ0 is counted not from the stress overshoot value but from the mechanical stress σM. The equation was used successfully to fit the long-time sigmoidal relaxations of Figures 5b and 6b at γ˘ ) 1.2 and 2 s-1, respectively. Just above the transition rate γ˘ I/N, we determine τNG ) 160 s with an initial amplitude of 17 Pa, while in the 2 s-1 spectrum, τNG is 20 s for an amplitude ∆σ0 of 26 Pa. In both cases, we take R ) 2. Note that the τNG variation as a function of shear rate is extremely rapid and then measurable in a restricted γ˘ range. Actually, τNG(γ˘ ) was related to a typical time featuring both the nucleation and the growth processes of the oriented phase in coexistence with the isotropic one. Since the transition is induced by shear, the growth of the nematic nuclei is expected to be highly anisotropic. Therefore, it was assumed that the rate-limiting process of growth operates essentially in the direction of the velocity gradient. This assumption implies that the dynamics of growth is a one-dimensional process, which gives rise to a stretched exponential decay with R ) 2 in the domain of metastability. According to Avrami and Kolgomorov’s rule,18 we are reminded that the time dependence of the already transformed volume fraction in the dynamics of a first-order transition goes as {1 exp(-td+1)}, where d is the dimension of homogeneous growth. In conclusion to this section, the long-time kinetics revealed by the transient rheology data can be ascribed (18) Delcourt, O.; Descamps, M.; Hilhorst, H. J. Ferroelectrics 1991, 124, 109-114.
Transient Rheology of Wormlike Micelles
to a transition from a flow that is homogeneous shortly after inception of the shear, say for t ∼ tM (but in a metastable state), and that becomes inhomogeneous as time evolves. It is of the utmost importance to keep in mind that the discontinuity of slope in the σST(γ˘ ) behavior at γ˘ I/N, the long-time transient kinetics, and the inhomogeneous shear flow are three features of the same physical phenomenon. 4.4. Plateau Regime: γ3 τR ∼ 1 and γ3 > γ3 M. We come now to the crucial issue of the paper and are interested in the γ˘ range where damped oscillations are controlling the transient response. This is the case above 2 s-1. As illustrated in Figures 6 and 7, the oscillations are merging progressively in the stress response as strain rate is increased. It is also important to notice that they coincide with the apparent maximum of the σM(γ˘ ) behavior (Figure 8), a point which will be commented on later. The period of the oscillating stresses has been estimated and found to vary very slowly in the γ˘ range investigated. Between 2 and 10 s-1, this period is 4 ( 1 s. It should finally be mentioned that, in this strain rate range, the micellar solution is exposed to extremely high stresses shortly after inception of the shear (t ∼ tOS); e.g., at 5 s-1, this is more than 1500 Pa! Oscillating stresses in start-up experiments are clearly not a usual feature for complex fluids under shear. On surfactant micellar solutions, however, this kind of behavior has already been emphasized by Shikata et al.14 eight years ago. The system investigated by these authors was again CTAB-NaSal/W. Ranges in surfactant concentrations, time scale, and shear rate were comparable to those of the present study. The interpretation however was formulated differently.14 Of major interest is that, first, the CTAB-NaSal/W surfactant system studied by Shikata et al. turned out to be a Maxwell fluid too with a unique terminal time τR ∼ 4-5 s. Second, the period they obtained (∼20 s) was compatible with that of the CPCl/Sal system if this latter one is expressed in units of τR. So, in both examples, CTAB-NaSal/W and CPCl/Sal, one gets a period of oscillations of about 4τR. According to us, the oscillations revealed in the transient stresses of wormlike micelles are the signature of a purely mechanical instability, e.g. as one of the type that Cates and co-workers have worked out.2,19,20 This mechanical instability is associated with the nonmonotonic σ(γ˘ ) behavior (Figure 1). This is equivalent to say that at some characteristic rate the local viscosity as defined by dσ/dγ˘ becomes negative.2,16 For the wormlike system, this is mechanical stress σM, which has to be considered (and not the stationary stress σST) in the treatment of the mechanical instability. σM represents the value of the stress in the material just before the solution phase separates (i.e. before the sigmoidal kinetics). In this respect, the σM maximum observed at 2 s-1 in Figure 8 should compare with the spinodal point M designated in Figure 1. However, because of the oscillations, σM becomes difficult to determine accurately at higher strain rates. We nonetheless interpret the last data points on the σM(γ˘ ) curve as an indication of the decrease of the mechanical stress at higher γ˘ and also as an evidence of the instability (dσM/dγ˘ < 0). The above assignment deserves some more comments. Recently, Spenley et al. have extended their calculations to the transient shear response using the original constitutive scheme of ref 20. As a result, they predict stress overshoots in start-up flow at all strain rates above 2/τR. (19) Callaghan, P. T.; Cates, M. E.; Rofe, C. J.; Smeulders, J. B. A. F. H. J. Phys. II, in press. (20) Spenley, N. A.; Yuan, X. F.; Cates, M. E. J. Phys. II 1996, 6, 551.
Langmuir, Vol. 13, No. 8, 1997 2233
Figure 10. Transient shear stress as calculated from the Johnson-Segalman model. With the set of constants used in eq 4 (τ ) 1 s, b ) 1/x2), σ(γ˘ ) has a maximum at γ˘ ) 1/τb ) x2 s-1. Note that, beyond this maximum (for γ˘ ) 2 and 5 s-1), the transients exhibit oscillations very similar to those of the wormlike micelles. However, the agreement is qualitative only.
These are qualitatively very similar to the ones shown in Figures 4-7. However, the overshoots persist in the unstable regime (dotted line in Figure 1) and are never followed by damped oscillations. In that sense, the integral constitutive equation developed for wormlike micelles2 deviates markedly from the rheological data obtained on CPCl/Sal at φ ) 12%. Better agreement can be foundsat least qualitativelysif one considers the phenomenological approach of Johnson and Segalman (J-S). The J-S model describes a nonaffine constitutive equation adapted for polymer solutions and melts.12 In addition to the prediction of a nonmonotonic flow curve similar to that of Figure 1 (assuming a Newtonian background), oscillations are found in the startup stress above the spinodal point.12 These aspects have been treated in detail by Greco and Ball, who had recently addressed the problem of the band formation and inhomogeneous Couette flow of wormlike micelles.21 We are here interested in the trivial case, that is, cone-and-plate geometry and strain-imposed rheometry. In this case, the stress response can be calculated analytically and following Larson,12 one gets
σ(t,γ˘ ) e-t/τ ) 1 - e-t/τ cos(γ˘ bt) sin(γ˘ bt) γ˘ bτ σ(tf∞,γ˘ )
(4)
where b ) (1 - a2)1/2 and a is the coefficient of slippage defined in the J-S model. In eq 4, a unique relaxation time τ has been assumed. Figure 10 displays the results of eq 4 for different strain rates below and above 1/τ. Here τ ) 1 s, b ) 0.707, and γ˘ is varied in the range 0.1-5 s-1. The qualitative agreement with the transient stress data obtained on the CPCl/Sal system is interesting. The exponential increase of the stress at low γ˘ develops into an overshoot, which in turn is progressively replaced by the damped oscillations (Figure 10). The oscillations coincide with the decrease of σ(γ˘ ) versus γ˘ . It should be noticed that direct fitting of the σ(t) transients using eq 4 has been attempted and that the agreement was rather poor. Actually, one obvious drawback of the J-S model relies on the fact that it predicts periods of oscillation T which are constant when expressed in terms in strain units (γ˘ T ) 2π/b). In other words, the period of the damped oscillations varies as 1/γ˘ . This is clearly not what was (21) Greco, F.; Ball, R. C. To appear in J. Non-Newtonian Fluid Mech.
2234 Langmuir, Vol. 13, No. 8, 1997
observed experimentally (we found out a period which varies very slowly with γ˘ ). 5. Concluding Remarks The transient rheology applied to wormlike micelles enables a very accurate experimental determination of their flow mechanisms. Two basic processes have been evidenced from start-up experiments. (1) The first mechanism occurs typically on the time scale of τR, the (unique) terminal relaxation time of the Maxwell viscoelastic fluid. In this time range (which can extend up to 10τR), the purely mechanical response of the micellar fluid is observed. Remarkably, it exhibits strong similarities with that of conventional polymers: elastic limit at γ , 1, stress overshoot around τR, and damped oscillations at higher strain rates. (2) The second basic mechanism is a long-time sigmoidal relaxation that was already identified8,9 with a transition from an homogeneous to an inhomogeneous shear flow (coexistence of phases of different viscosity, orientation, and order parameters). The fundamental feature of this nonequilibrium transition is a constant stress plateau (without hysteresis). Of major relevance, the dynamics of the shear-induced phase transition occurs on a totally different time scale than that of the mechanical response: t . τR. From the present transient rheology, we have been able to extract the purely mechanical response of the wormlike micelles. The so-called mechanical stress σM has been shown to pass through a maximum at γ˘ M, as one would have expected for a purely mechanical instability (decreasing branch of σ(γ˘ ) in Figure 1). The damped oscillating stress evidenced at the maximum (and above γ˘ M) is also a crucial indication that the flow is becoming unstable. This suggests that, in wormlike micellar solution, the purely mechanical instability as followed by the σM(γ˘ ) flow curve is preempted by the transition toward
Berret
an (nematic) inhomogeneous flow. Both mechanisms are however intertwined. Expressed differently, the crossover toward the plateau regime (γ˘ ) γ˘ I/N) occurs before the spinodal point shown by the mechanical response (γ˘ ) γ˘ M). Underlying the present description of the shearinduced transition in wormlike micelles, there is the formal analogy with the quenching of an equilibrium system into (1) a metastable regime characterized by nucleation and growth and (2) an unstable regime of spinodal decomposition. Caution should be supplied in this comparison, since we are here dealing with nonequilibrium systems. This interpretation of the nonlinear rheology of wormlike micelles has been used as the basis of a quasithermodynamic approach recently developed by Porte and co-workers.22 The main issue of this model was to introduce an elastic effective potential which in analogy with equilibrium thermodynamics could be minimized, yielding a unique and robust path in the stress behavior between the low- and high-strain-rate branches (the plateau at σ ) σI/N, see Figure 2). As a strong conclusion, this suggests that a nonmonotonic mechanical stress behavior is the prerequisite for the wormlike system to undergo a transition toward an inhomogeneous flow. Acknowledgment. During the course of this work, I have benefited from incalculable discussions with Gregoire Porte that I would like to acknowledge here. I am also indebted to F. Greco, who pointed out the qualitative analogy with the Johnson-Segalman model. J. Harden, F. Molino, and P. Olmsted are also acknowledged for fruitful discussions. This work has benefited from the stimulating environment provided by the GDR 1081 “Rhe´ophysique des Colloı¨des et Suspensions” founded by the CNRS. LA961078P (22) Porte, G.; Berret, J.-F.; Harden, J. H. J. Phys. II, in press.
