Transient Rheology of a Polymeric Bicontinuous Microemulsion

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Transient Rheology of a Polymeric Bicontinuous Microemulsion Kasiraman Krishnan,† Wesley R. Burghardt,‡ Timothy P. Lodge,*,†,§ and Frank S. Bates*,† Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, Department of Chemical Engineering, Northwestern University, Evanston, Illinois 60208, and Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 Received June 17, 2002. In Final Form: September 3, 2002 Transient rheological measurements are reported for a model polymeric bicontinuous microemulsion. The sample consists of a ternary blend of poly(ethyl ethylene) (PEE) and poly(dimethyl siloxane) (PDMS) homopolymers and a symmetric PEE-PDMS diblock copolymer. Steady-flow rheological data, reported previously, show four regimes as a function of increasing shear rate. Newtonian behavior is observed in regime I, followed by shear thinning in regime II. Flow-induced phase separation is the hallmark of regime III. The microemulsion starts ejecting homopolymer-rich phases, and the shear stress is independent of shear rate. In regime IV, complete phase separation occurs and the sample behaves like an immiscible blend. Transient rheological data on flow inception reveal linear viscoelastic response in regime I, and development of a stress overshoot in regime II. In regime III, a strong stress overshoot is observed, followed by a “shoulder” and a slow decay to the steady, rate-independent value. The normal stress shows a similar response as the shear stress. The transient morphology development has been further characterized using light scattering and microscopy. A “double-streak” pattern develops in light scattering on flow inception, which ultimately evolves into a “single-streak” pattern. Optical microscopy shows development of a stringlike morphology, indicating a transition in lengthscale from tens of nanometers to microns. Transient rheological measurements on step changes in shear rate, both within and between regimes III and IV, and also on flow cessation, support the proposed morphologies.

1. Introduction Ternary polymer blends comprising two homopolymers and the corresponding diblock copolymer resemble oil/ water/surfactant systems1 and exhibit similar self-assembled morphologies. A bicontinuous microemulsion2 is one such morphology, consisting of co-continuous domains of the two polymers and the copolymer surfactant lying at the interface.3,4 This morphology is characterized by extensive interfacial area and vanishingly small interfacial tension. Polymeric bicontinuous microemulsions occur over a narrow window of amphiphile concentration, between a multiphase region and a swollen lamellar phase. Unlike the oil/water/surfactant systems, however, this microemulsion window spans a wide range of temperature. The phase behavior of these ternary polymer blends has already been explored in detail.1,3-5 We also recently demonstrated the richness of the rheological response of a polymeric bicontinuous microemulsion6 formed in a ternary blend of poly(ethyl ethylene) (PEE), poly(dimethyl * Corresponding authors. † Department of Chemical Engineering and Materials Science, University of Minnesota. ‡ Department of Chemical Engineering, Northwestern University. § Department of Chemistry, University of Minnesota. (1) Washburn, N. R.; Lodge, T. P.; Bates, F. S. J. Phys. Chem. B 2000, 104, 6987-6997. (2) Scriven, L. E. Nature 1976, 263, 123-125. (3) Bates, F. S.; Maurer, W. W.; Lipic, P. M.; Hillmyer, M. A.; Almdal, K. S.; Mortensen, K.; Fredrickson, G. H.; Lodge, T. P. Phys. Rev. Lett. 1997, 79, 849-852. (4) Hillmyer, M. A.; Maurer, W. W.; Lodge, T. P.; Bates, F. S.; Almdal, K. J. Phys. Chem. B 1999, 103, 4814-4824. (5) Kielhorn, L.; Muthukumar, M. J. Chem. Phys. 1997, 107, 55885608.

siloxane) (PDMS) and a PEE-PDMS diblock copolymer. Among other attributes, this system exhibited a stress plateau as a function of shear rate in steady shear flow. The rheology and structure of this sample under equilibrium and steady-state conditions have been presented in previous publications.6,7 The present study addresses the issues of transient behavior on the startup and cessation of steady shear, and on step changes in shear rate. The bicontinuous PEE-PDMS microemulsion exhibits four regimes as a function of shear rate in steady shear flow.6,7 Figure 1 depicts the key results from rheological measurements. Figure 1a shows the stress, Figure 1b the first normal stress difference (N1), and Figure 1c the viscosity of the bicontinuous microemulsion and the constituent homopolymers, as functions of shear rate. At very low shear rates (regime I) the sample is a Newtonian fluid, whereas shear thinning is observed at intermediate rates (regime II). Scattering experiments indicate development of anisotropy in the bicontinuous structure within regime II. Upon entering regime III, flow-induced phase separation ensues, and the microemulsion ejects two homopolymer-rich phases. We proposed a three-phase coexistence for this regime: the bicontinuous microemulsion coexisting with two homopolymer-rich phases. The significant feature of this regime is that both the shear stress and normal force are independent of shear rate. In situ optical microscopy reveals a micron-scale stringlike morphology, whereas light scattering shows a streaklike (6) Krishnan, K.; Almdal, K.; Burghardt, W. R.; Lodge, T. P.; Bates, F. S. Phys. Rev. Lett. 2001, 87, 098301/1-4. (7) Krishnan, K.; Chapman, B. R.; Bates, F. S.; Lodge, T. P.; Almdal, K.; Burghardt, W. R. J. Rheol. 2002, 46, 529-554.

10.1021/la026081p CCC: $22.00 © 2002 American Chemical Society Published on Web 11/09/2002

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The stress plateau seen in this system is reminiscent of those reported in wormlike micellar surfactants8-12 in recent years. Despite inherent differences between the two systems, we will show that there are remarkable similarities in the rheological behavior. In the wormlike micelles, stress plateaus have been interpreted in two different ways. One explanation postulates an underlying nonmonotonic dependence of stress on shear rate in a single fluid phase, leading to a shear banding instability.9 Within the stress plateau, bands of low-viscosity fluid sustaining a higher shear rate coexist with bands of highviscosity fluid sustaining a lower rate. The second explanation10-12 postulates a shear-induced first-order phase transition between isotropic and nematic phases, where the stress plateau spans the biphasic region. In some wormlike surfactant solutions, in situ neutron scattering provides direct evidence of a shear-induced nematic phase. However, because the low-viscosity branch of the flow curve should be associated with a highly aligned state, according to tube-model-based interpretations of the shear-banding phenomenon, the distinction between these two interpretations of stress plateaus in surfactants is clouded. In the PEE-PDMS microemulsion,6,7 there is direct evidence of a bulk-phase separation associated with the shear stress plateau, most convincingly manifested in easily observable dispersed-phase droplets on cessation of shear, which require tens of minutes to redissolve into the single-phase microemulsion. At the same time, it is intriguing that the one existing theory of bicontinuous microemulsion rheology (based on a nonequilibrium Landau-Ginzburg model) predicts a nonmonotonic dependence of shear stress on shear rate for the excess viscosity of the microemulsion,13 and hence is also presumably capable of predicting shear-banding instabilities. Berret and co-workers14,15 have investigated the transient rheological behavior of wormlike micelles during the startup of steady shear. Three time ranges were identified: a very short time elastic solidlike response, an intermediate timescale stress overshoot as in conventional polymers, and a long time response leading to inhomogeneous flow. The transient rheological response of the bicontinuous microemulsion, reported here, resembles this in certain ways. The present paper also offers a systematic comparison between the two systems, and invokes analogies with other categories of soft materials. 2. Experiments Figure 1. Steady shear rheological data at 15 °C. (a) Shear stress vs shear rate with increasing (9) and decreasing (O) rates. Four regimes are observed as a function of shear rate and no hysteresis occurs in steady-state values. (b) First normal stress difference vs shear rate. Open squares are the data collected using ARES and the solid squares are those obtained with RFS; / indicate data in reverse direction obtained with ARES. (c) Viscosity of the bicontinuous microemulsion (9) along with that of PEE (4) and PDMS (/) homopolymers as functions of shear rate.

2.1. Materials. The bicontinuous microemulsion used for the present study consists of a ternary blend of PEE, PDMS, and the diblock copolymer PEE-PDMS. All the polymers were synthesized by anionic polymerization techniques as described by Hillmyer et al.4 PEE was prepared by deuterating 1,2-polybutadiene in both the homopolymer and the diblock copolymer samples, thereby enhancing small-angle neutron scattering (SANS) contrast. The number-average molecular weights as determined by NMR were 1770, 2130 and 10 400 for PEE, PDMS, and PEEPDMS, respectively. Densities were measured using a density

pattern. At very high rates (regime IV), complete phase separation occurs and the sample responds similarly to an immiscible binary polymer blend; here too the stress increases with shear rate. The hypotheses about these regimes are also supported by the rheological response of the constituent homopolymers as shown in Figure 1c, where the various viscosities are compared. The system goes into phase separation because ejection of the lowviscosity PDMS-rich phase provides an effective stresslimiting mechanism.

(8) Rehage, H.; Hoffmann, H. J. J. Phys. Chem. 1988, 92, 47124719. (9) Spenley, N. A.; Cates, M. E.; Mcleish, T. C. B. Phys. Rev. Lett. 1993, 71, 939-942. (10) Olmsted, P. D.; Lu, C.-Y. D. Phys. Rev. E 1997, 56, R55-R58. (11) Olmsted, P. D. Europhys. Lett. 1999, 48, 339-345. (12) Berret, J.-F.; Roux, D. C.; Lindner, P. Eur. Phys. J. B 1998, 5, 67-77. (13) Pa¨tzold, G.; Dawson, K. Phys. Rev. E 1996, 54, 1669. (14) Berret, J.-F. Langmuir 1997, 13, 2227-2234. (15) Lerouge, S.; Decruppe, J.-P.; Berret, J.-F. Langmuir 2000, 16, 6464-6474.

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gradient column, and the volume fraction of PEE in the block copolymer was determined to be 0.52. The low molecular weights facilitate accessing the high-temperature disordered regime, and reduce kinetic barriers for reaching thermodynamic equilibrium. The block copolymer molecular weight was chosen to be about five times that of the homopolymers so that the order-disorder transition (90 °C) of the former is close to the critical temperature of the latter (150 °C). All experiments were performed on a single bicontinuous microemulsion sample containing 10% block copolymer and equal amounts of homopolymers by volume. The blend was prepared by direct mixing at 80 °C (in the disordered state) and then cooled to room temperature, at which it forms a single-phase bicontinuous microemulsion. This microemulsion has a domain periodicity of 75 nm and a correlation length of 35 nm at room temperature, as obtained from SANS measurements. 2.2. Experimental Techniques. Rheological measurements were performed with two Rheometric Scientific strain-controlled rheometers (ARES and RFS). A cone-and-plate geometry was used with 50-mm-diameter fixtures and either a 0.02 (ARES) or 0.04 (RFS) radian cone angle. The sample temperature was controlled to within (0.1 °C by a convection oven in the ARES; the oven also provided a blanket of nitrogen. Temperature was controlled with a water-circulation bath in the RFS, and was maintained to within (0.2 °C. All shear stress measurements were performed with the ARES, whereas normal stress measurements at low shear rates used the more sensitive RFS. The cone-to-plate gap was kept constant by accounting for the thermal expansion coefficient of the tools. The sample was viewed at high magnification with a video camera to verify the absence of edge fracture at high-shear rates. Transient experiments were conducted for startup of shear flow in each of the four regimes. In addition, transient measurements were made for jumps in shear rate across various regimes, both with increasing and decreasing rates. Stress relaxation data also were collected after cessation of steady shear. Time-resolved experiments were conducted using the flowlight-scattering apparatus described in detail in Krishnan et al.7 The setup contains a shear stage (Linkam Scientific Instruments) using a parallel plate geometry with transparent disks. The sample thickness was between 0.5 and 0.8 mm. Light from a He-Ne laser source (632.8 nm) was directed perpendicular to the disks and the scattering patterns were projected onto a ground-glass screen placed on the other side. The images were captured using a CCD camera connected to a VCR and a computer with frame-grabbing capability. Time-resolved patterns were obtained by recording the sequence of scattering events on highquality S-VHS videotapes using a professional grade VCR, and by grabbing the images later with appropriate software. Real space images were obtained by mounting the same shear stage on an optical microscope (Olympus) with a long workingdistance objective. The light-scattering and microscopy observations are performed only at a single radial position in the parallel plate shear cell. Comparisons are made with the rheological data obtained with cone-and-plate geometry, at the shear rate corresponding to this radial position. The discrepancies due to flow geometry are minimal, because the lengthscale of the structures are of the order of microns, which is small compared with the dimensions of the shear cell.

3. Results: Shear Flow Inception 3.1. Rheology. Steady shear experiments revealed four regimes as a function of shear rate as discussed before, and the first normal stress difference (Figure 1b) exhibits almost the same overall response as the shear stress (Figure 1a). The results from shear flow startup experiments are presented in Figures 2 and 3. Parts a and b of Figure 2 show the viscosity and shear stress, respectively, as a function of time, at different shear rates. The strain at which the first overshoot in stress occurs is plotted against shear rate in Figure 2c. Within regime I, the shear stress and viscosity increase monotonically to the steady value on inception of steady shear; no overshoot or undershoot is observed. Data at multiple shear rates within regime I (Figure 1c) reveal Newtonian behavior at

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Figure 2. Transient rheological data on the startup of steady shear at 15 °C. (a) Viscosity and (b) shear stress vs time at the following shear rates: Regime I, 0.05 s-1 (0); regime II, 0.316 (O), 0.7 (]), and 1.0 (3) s-1; regime III, 2.0 (+), 3.16 (4), and 5.0 (×) s-1. The solid line in panel (a) represents the fit obtained using independently measured linear viscoelastic spectrum.7 (c) Shear rate dependence of the strain corresponding to the initial overshoot.

steady state, whereas the transient viscosity growth may be well described using the independently measured linear viscoelastic spectrum.7 On entering regime II, overshoots are observed in both the stress and viscosity. The overshoots become more pronounced with increasing rate. Comparison of steady-state values of the viscosity at different rates (Figure 1c) reveals the steady-state shear thinning. The transient behavior becomes more complicated in regime III. Figure 2 shows representative data at a few shear rates. On the inception of steady shear, an initial sharp overshoot occurs, resembling those observed in


Figure 3. Transient first normal stress difference as a function of time at 15 °C, at the following shear rates: 0.316 s-1 (O), 0.5 (0), 1.0 (1), 2.0 (+), 3.16 (4), and 5.0 (×) s-1.

regime II. The stress/viscosity quickly reaches a plateau, but thereafter slowly decreases to the final steady-state value. The plateau or the “shoulder” is very distinct at low shear rates within regime III, but its duration decreases with increasing rate. At higher rates, a broad undershoot also becomes apparent after the shoulder as the stress approaches its final value. These data are reproducible, and the same phenomena are observed at different temperatures. The most important feature is that the steady-state value of the stress is almost independent of shear rate in this regime (Figure 2b). Flowinception experiments at rates within regime IV were hindered by edge fracture. This seems to be associated with the very high overshoot stresses generated during the transient period, because edge fracture does not occur when the shear rate is progressively increased stepwise into regime IV (as is done for steady-state measurements). The first overshoot in stress occurs at ∼2 strain units at low rates, and at progressively greater strain units for higher rates. More discussion of this follows. Transient normal stress data are shown in Figure 3. Within regime III, the first normal stress difference exhibits behavior similar to the shear stress: an initial overshoot followed by a shoulder and a slow decay to steady state. Although the data are noisy, it seems that the steadystate value of the normal force in regime III is independent of shear rate, similar to the shear stress. 3.2. Flow Light Scattering. Flow-light-scattering experiments were conducted at different shear rates and temperatures. At low-shear rates (regimes I and II), very little scattering ocurs, because the lengthscale of the microemulsion structure is too small to be discerned. However, in regimes III and IV, a bright streak pattern emerges perpendicular to the flow direction. The steadystate light-scattering patterns have been described in detail previously7; this scattering is associated with a shear-induced phase separation in this sample. The transient phenomena leading to flow-induced phase separation, and hence the streak-like pattern, were investigated in detail. Figure 4 shows representative results from these time-resolved light-scattering experiments in regime III. The scattering patterns at different times are presented, along with the corresponding real space images and the rheological response. In the quiescent state only stray scattering near the beamstop is observed. On inception of shear flow, and only after completion of the initial ‘fast’ stress overshoot, a butterfly-like pattern develops. The wings of the butterfly gradually approach

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each other, ultimately merging into a bright streak perpendicular to the flow direction. As indicated in Figure 4, this evolution in light scattering occurs across precisely the same timescale as the ‘slow’ process by which the stress drops to its steady value. The steady-state bright streak pattern increases in intensity with increasing shear rates across regime III. The time to reach the steady state also decreases with increasing rates, in accord with the transient rheological data in Figures 2 and 3. These experiments were repeated at different temperatures from 10 to 30 °C, and similar results were obtained. The overall intensity of the scattering patterns was higher at lower temperatures. The evolution of the bright streak was also slower at lower temperatures. The critical shear rate for the onset of the streaklike pattern increased weakly with temperature, and is well correlated with critical shear rate for onset of the stress plateau in the steady-state rheology. 3.3. Optical Microscopy. The quiescent sample appears homogeneous and clear under an optical microscope (except for a few dust particles). The real space images remain unchanged after the inception of shear flow when the shear rates are in regimes I and II. However, in regime III, an overall darkening of the image occurs, followed by the development of grainlike structures that elongate with time, ultimately leading to a stringlike morphology at steady state. Figure 4 shows these results at 23 °C along with the complementary light-scattering and rheology data. The steady-state morphologies at different shear rates have been presented earlier,7 which show that the number of strings increase with shear rate, accompanied by an increased sample turbidity. The stringlike morphology also develops sooner at higher rates. Similar phenomena were observed at other temperatures, with slower dynamics at lower temperatures. 3.4. Analysis and Discussion. Rheological studies on the bicontinuous microemulsion show a systematic variation in the transient behavior as a function of shear rate (Figures 2 and 3). In regime I, i.e., the Newtonian regime, the sample exhibits a typical linear viscoelastic response. At higher rates (regime II), development of a shear stress overshoot occurs. The behavior within regime II closely resembles the nonlinear viscoelasticity of conventional entangled polymers; so much so, in fact, that it is worth reminding the reader that the pure homopolymers in this sample are below their entanglement molecular weights and have immeasurably small elastic character at the deformation rates considered here. In this case, the nonlinear stress response is associated with the stretching and/or orienting of the bicontinuous domains due to flow. The instantaneous response on inception of shear reflects the linear response of the undeformed bicontinuous structure. However, as the domains become aligned with the flow, the viscosity decreases, leading to the overshoot behavior. Shear flow rotates interfaces initially aligned perpendicular to the flow direction, thereby reducing the degree of percolation along the velocity gradient directions the mechanism identified by Anklam et al.16 to explain shear thinning in oil/water/surfactant bicontinuous microemulsions at very high shear rates. The strong viscosity contrast between PEE and PDMS homopolymers ensures that such a progressive loss of continuity in the more highly viscous PEE domains has enormous potential for shear thinning (Figure 1c). Although the microscopic mechanism responsible for the nonlinear response in regime II is profoundly different (16) Anklam, M. R.; Prud’homme, R. K.; Warr, G. G. AIChE J. 1995, 41, 677-682.

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Figure 4. Time-resolved light-scattering and optical microscopy data during startup of steady shear (18.75 s-1, 23 °C) in regime III. The top plot shows the shear stress as a function of time and the instants (A-F) at which the light-scattering and microscopy images were obtained. The central dark circle in the scattering patterns arises from the beamstop. Flow direction is vertical; hence, the plane under investigation is the flow-vorticity plane.

from that active in entangled flexible polymers, there are noteworthy similarities in certain details. This is particularly evident in Figure 2c, which presents the shear strain (γp ) γ˘ tp) at which the shear stress overshoot peak appears as a function of shear rate. The peak strain is near 2 when the stress overshoot first appears, and then shifts toward larger values at higher rates. Both of these features closely resemble the behavior of entangled flexible polymers,17,18 as well as entangled wormlike micelle solutions.14 The fact that the stress overshoot initially appears at a strain of 2 is perhaps not surprising, because this factor is primarily geometric in origin. The affine rotation of an initially random ensemble of thin rods or platelets leads to an orientation distribution in which the shear component of the second moment tensor passes through a maximum at a strain of 2. It is plausible that the rotation of microemulsion interface on flow inception leads to similar geometric characteristics. Changes in the peak strain at higher shear rates are harder to rationalize. In entangled flexible polymers, such changes are believed to reflect the action of transient chain stretching.19 In the present microemulsion case, the homopolymers exhibit purely Newtonian rheology, hence the analogous process could be the stretching of the domains as a whole. We also note that an anomaly exists in the microemulsion shear stress behavior during inception that has no analogues in flexible polymers; this will be discussed later. (17) Menezes, E. V.; Graessley, W. W. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 1817-1833. (18) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986. (19) Pearson, D.; Herbolzheimer, E.; Grizzuti, N.; Marrucci, G. J. Polym. Sci. B: Polym. Phys. 1991, 29, 1589-1597.

Stress overshoots occur in entangled flexible polymers when the product of the shear rate and the longest relaxation time, γ˘ τ, is greater than unity.17 In a previous publication,7 we reported the characteristic relaxation time of the bicontinuous microemulsion as 3.4 s at 15 °C as measured by dynamic mechanical spectroscopy (DMS) experiments. This indicates that nonlinearity should occur above ∼0.3 s-1, which agrees reasonably with the onset of overshoot in the transient experiments. Flow-induced phase separation commences in regime III and leads to (i) a plateau in the steady-state stress and (ii) more complex transient behavior, described in Section 3.1. We have explained this stress-shear rate plateau6,7 in terms of increasing degree of phase separation. Specifically, we have proposed a three-phase coexistence in this regime, in which the bicontinuous microemulsion expels increasing amounts of homopolymer-rich phases with increasing rate, maintaining the stress at a constant level. The stress level obtained during the intermediate stage of flow inception in regime III provides further insight into the phenomenon of flow-induced phase separation. Figure 5 presents steady-state stress and viscosity data onto which solid symbols representing the ‘shoulder’ stress/ viscosity values are added. (The shoulder values are extracted from transient data such as in Figure 2.) Focusing on the stress data for 15 °C, there is initially a linear dependence of stress on shear rate in regime I (slope ) 1), followed by the onset of shear thinning in regime II (slope ) power law index ) 0.6). Within regime III, Figure 5 illustrates that the metastable ‘shoulder’ stress levels seen during the flow inception line up as a nearly perfect extrapolation of the power-law shear thinning present in


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Figure 5. Viscosity corresponding to the “shoulder” in transient data (e.g., Figure 2), plotted along with the steady shear results at 10 °C (4), 15 °C (3), and 23 °C (0). The “shoulder” viscosities (solid symbols) line up as an extrapolation of the viscosity in regime II. The inset shows the steady-state and “shoulder” data of stress at 15 °C (/).

regime II. This implies that the system evolves temporarily through a morphology that is a simple extension of the behavior in regime II, before ultimately submitting to phase separation. Within regime II, we attribute shear thinning to development of a highly anisotropic, but still nanostructured and possibly bicontinuous morphology. When shear flow is imposed at a shear rate within regime III, the system first attains a stretched morphology that is similar to regime II. However, at such high rates, the stress that is being supported by this highly deformed structure apparently exceeds some critical value, and phase separation intervenes as a way to return the mixture to its maximum sustainable stress. Viscosity data presented in Figure 5 verify that this sequence of events occurs at various temperatures. In addition, they illustrate the remarkable fact, previously noted,7 that the viscosity in regime III (and hence the plateau stress) is almost independent of temperature. Transient rheological behavior similar to that in regime III has been reported in the wormlike micelles.15,20 The slow decay of stress toward an ultimate steady-state value has been modeled by using a sigmoidal profile, R

[ (τt) ]

∆σ ∼ exp -

(1)

where ∆σ is the extra stress corresponding to the shoulder relative to the ultimate steady-state value. This equation was derived using a nucleation and one-dimensional growth model in which there was a shear-induced anisotropic growth of a nematic phase from the isotropic phase. Although this model may not be appropriate in detail for our system because of the absence of shear banding, it fits reasonably well for shear rates just above the onset of phase separation, with the exponent R ∼ 2. For higher rates within the plateau, it fails because of the develop(20) Berret, J.-F.; Roux, D. C.; Porte, G. J. Phys. II France 1994, 4, 1261-1279.

ment of a distinct undershoot. A similar problem was also reported in the wormlike micelle case.15,20 In micelles, however, the transients can become considerably more complicated with pronounced stress oscillations that ultimately overwhelm the slow transient process.14 The origin of this undershoot in the microemulsion sample is not yet apparent. The normal stress data are also interesting. As depicted in Figure 1b, the first normal stress difference (N1) follows the same trends as the shear stress in regimes III and IV. In particular, N1 exhibits a plateau in regime III. This is in sharp contrast with the wormlike micelle case, where N1 exhibits a linear increase with shear rate, as predicted by models of shear banding.9 This strongly suggests that the stress plateau in the present case is fundamentally different than those associated with shear-banding instabilities in one-phase fluids. Conversely, this behavior is coincident with the notion of associating the stress plateau in the microemulsion with flow-induced phase separation and not shear banding. The transient data for normal stress also show the same response as the shear stress in regime III, i.e., an initial overshoot followed by a shoulder and then a slow decay to steady state. In regime II, however, the overshoot in normal stress is not as distinct as that in the shear stress. This behavior resembles entangled flexible polymers, where normal stress overshoots are typically known to occur at higher shear rates than the shear stress overshoot,18 attributed to chain-stretching effects. Time-resolved light-scattering experiments on the flowinduced phase separation in regime III reveal the formation of a “double-streak” pattern that transforms into a “single-streak” pattern. Streaklike patterns have been reported while shearing polymer blends21-24 and solutions.25 This kind of pattern arises because of a stringlike morphology. The q-range of the patterns also indicates that the lengthscale of the structures is of the order of microns. During the early stages of this process, the pattern somewhat resembles the butterfly patterns associated with shear-enhanced concentration fluctuations in semidilute polymer solutions.25 The concentration fluctuations, which here ultimately result in phase separation, are initially predominant in the flow direction, resulting in the butterfly pattern. The macroscopic phase-separated structures gradually build up, leading to diminution of fluctuations and suppression of intensity along the flow direction. The latter stages of the transient patterns are similar to the ones reported by Weiss and co-workers21,26 in the polymer blends. The evolution of the double-streak pattern into a single-streak pattern can be attributed to a progressive increase in the size of the phase-separated domains. The phase separation accompanied by stretching due to flow causes a significant difference between the lengthscales parallel and perpendicular to the flow direction. The lengthscale is inversely related to the position of the intensity maxima in either direction. (The position (21) Hong, Z.; Shaw, M. T.; Weiss, R. A. Polymer 2000, 41, 58955902. (22) Kielhorn, L.; Colby, R. H.; Han, C. C. Macromolecules 2000, 33, 2486-2496. (23) Fernandez, M. L.; Higgins, J. S.; Richardson, M. Polymer 1995, 36, 931-939. (24) Kim, S.; Hobbie, E. K.; Yu, J.-W.; Han, C. C. Macromolecules 1997, 30, 8245-8253. (25) Kume, T.; Hattori, T.; Hashimoto, T. Macromolecules 1997, 30, 427-434. (26) Chen, Z. J.; Shaw, M. T.; Weiss, R. A. Macromolecules 1995, 28, 648-650.

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Figure 6. Light-scattering data analysis. Intensity (in arbitrary units) along vertical slices at different q⊥ positions. Parts A-F directly correspond to the light-scattering patterns in Figure 5, at different times as indicated. The solid lines are smooth curves drawn for ease of visualization. (q|| and q⊥ are the scattering vectors parallel and perpendicular to the flow direction, respectively.) The inset in part c shows the q⊥ positions at which the scans were taken; the symbols, O, 4, 3, and ], in this order, represent increasing q⊥.

of the intensity maximum perpendicular to the flow direction apparently goes out of range of detection capability.) The anisotropy in the lengthscale increases, and the two streaks come closer and closer to each other, ultimately resulting in the single-streak pattern. At this point it is worth noting that the bicontinuous microemulsion is peculiar in the sense that it shows characteristics common with semidilute solution butterflylike scattering, as well as the streaklike scattering of stringy multiphase fluids. Wormlike micelles somewhat resemble this, where a butterfly pattern is seen on shear flow inception and a bright streak is obtained at long times after shear banding.27 Figure 6 shows the intensity along vertical slices at different q positions, corresponding to the scattering

patterns in Figure 4. The two peaks correspond to the two streaks of the double-streak pattern. In Figure 6b and c, these two peaks do not seem to have a tendency to merge even at large q values. However, in Figure 6d-f, the two peaks seem to ultimately merge with increasing q. These results are similar to those of Hong et al.,21 the former is similar to their “converging” and the latter to the “diverging” pattern. The converging pattern arises because of the progressive increase in lengthscale as described above. Hong et al.21 attribute the diverging pattern to a chevronlike morphology formed due to flow instabilities. Because of flow instabilities there is a component of (27) Decruppe, J. P.; Lerouge, S.; Berret, J.-F. Phys. Rev. E 2001, 63, 022501.


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velocity in the radial direction, resulting in a helical flow. The projection in the flow-vorticity plane is a zigzag, chevronlike structure, which gives rise to the “diverging” pattern, where the two streaks do not touch each other even at high q. It is not clear how much of this argument based on polymer blends can be applied to the present system. However, it is certain the latter stages, i.e., the converging patterns, correspond to growth in lengthscale accompanied by increasing anisotropy. 4. Results: Step Changes in Shear Rate To further investigate the dynamics of the flow-induced phase separation within the stress plateau, transient rheological data were obtained for step changes in shear rate across regime III. Parts a and b of Figure 7 depict the results for increasing and decreasing steps in shear rate, respectively. After an increase in shear rate, the stress shows an initial overshoot followed by a slow decay with a distinct shoulder. The transient stress response is very similar to that observed on startup from the quiescent state, suggesting that the mechanism of morphology development is also similar. On a jump in shear rate, there is initial deformation of the microemulsion component of the blend (during the initial overshoot), which increases the stress above the critical value described before. As with flow inception, the system can return to this stress value via further expulsion of homopolymer-rich phases on longer timescales, leading to the slow stress decay back to the plateau. When a step decrease in shear rate occurs, the stress immediately decreases. However, the system finds itself in a phase-separated state in which there is an excess of homopolymer-rich phases (and, in particular, low-viscosity PDMS), and realizes that it is capable of supporting a higher stress at the lower rate. Thus, a part of the phaseseparated blend reverts back to the thermodynamically preferred bicontinuous nanostructure, reflected in a slow increase in the stress, which again reaches the plateau level. As discussed before with reference to Figure 1a, the plateau value of the shear stress is the same both with increasing and decreasing rates, thus establishing the robustness of stress plateau. Cappelaere et al.28 have reported a nondegenerate stress plateau devoid of any hysteresis in the wormlike micelles supporting their argument of phase transition. The microemulsion shows similar features. Figure 8 presents shear stress data for step changes in shear rate within regime IV. In this regime, the system is already completely phase separated, and seems to behave essentially like a conventional immiscible blend under shear. On a step increase in the rate, there is no longer a strong stress overshoot caused by the transient deformation of the microemulsion; instead, only a small overshoot is observed which probably reflects transient deformation of the phase-separated droplets.29,30 Conversely, for a step decrease in shear rate, no significant slow decay mode exists as in regime III. Instead, a rapid undershoot/overshoot combination occurs, followed by a very small secondary undershoot as the stress approaches steady state. The lack of a pronounced slow undershoot is consistent with the proposed morphology within regime IV, which is devoid of any microemulsion phase. (28) Cappelaere, E.; Berret, J.-F.; Decruppe, J. P.; Cressely, R.; Lindner, P. Phys. Rev. E 1997, 56, 1869-1878. (29) Minale, M.; Mewis, J.; Moldenaers, P. AIChE J. 1998, 44, 943950. (30) Vinckier, I.; Moldenaers, P.; Mewis, J. J. Rheol. 1997, 41, 705718.

Figure 7. Transient rheological data showing stress (O) vs time for step changes in shear rate across regime III for (a) increasing and (b) decreasing rates at 15 °C. The shear rates are indicated with solid lines, using the right ordinate.

Figure 8. Transient rheological data showing stress (O) vs time for a step increase and a step decrease in shear rate within regime IV, at 15 °C. The shear rates are indicated with solid lines, using the right ordinate.

The transient stress data (Figure 9) for jumps in shear rate between regimes III and IV are more interesting. When the shear rate is increased from regime III to IV (5 s-1 to 50 s-1), a strong overshoot occurs. There is rapid onset of complete phase separation, as the stress level that has to be sustained becomes very high. This shows up as the second decay in the stress to the steady value. (Note that this steady value is above the regime III plateau value.) When the shear rate is stepped down from regime IV to III (50 s-1 to 5 s-1), the short time response shows an undershoot/overshoot combination similar to step-down experiments within regime IV. Subsequently, a secondary

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Figure 9. Transient rheological data of the microemulsion (O) for a step change in shear rate from regime III (5 s-1) to regime IV (50 s-1) and back (15 °C). Transient data for glycerin (0) under identical conditions is also shown in the same graph. The shear rates are indicated with solid lines, using the right ordinate.

Figure 10. Stress relaxation for flow cessation after steady shear at 0.1 (0), 0.7 (O), 5 (4), and 50 s-1 (3) (T ) 15 °C).

undershoot occurs followed by a very slow rise of the stress. This is associated with the mixing of a large part of the phase-separated structures to recover most of the microemulsion, resulting in the three-phase state corresponding to that particular shear rate in regime III. The origin of the overshoot immediately on stepdown of shear, both within regime IV and from regime IV to III is unclear. Flow-cessation experiments within regime IV show only a monotonic decay of stress (Figure 10). A control experiment was performed with glycerin, which is a Newtonian fluid of comparable viscosity, with the same shear rates. The data for glycerin are also presented in Figure 9; no overshoot or undershoot occurs. Hence, it is clear that this complicated rheological response arises solely from the rapidly evolving microstructure and not because of inertial effects or any other experimental artifacts. The mechanism of recovery after cessation of shear from regime IV was discussed previously,6 the salient feature being a rapid development of almost isotropic spinodal-like structures. We speculate that a stepdown in shear rate leads to partial development of similar structures that are stretched because of continued flow, leading to an overshoot. The same phenomenon occurs for stepdowns both within regime IV and from regime IV to III; however, the slower process of reformation of the majority of the microemulsion phase is superimposed in the latter case.

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The stress relaxation after cessation of steady shear was also examined. Figure 10 shows representative data for cessation from steady shearing in different regimes. At the lowest rate, the relaxation data approach the predictions of linear viscoelasticity, with use of the independently measured relaxation spectrum. The stress relaxation follows similar timescales with a smooth decay in regimes I and II, where the morphology is still bicontinuous. In regime III, where homopolymer-rich phases occur in addition to the bicontinuous microemulsion, two distinct relaxation modes can be seen. The longterm relaxation behavior is similar to regimes I and II; however, a distinct fast mode occurs in the beginning, coincident with the retraction of elongated phaseseparated domains into droplets. The scenario becomes much more drastic in regime IV; at these high rates, only homopolymer-rich phases occur, and the stress relaxes much more rapidly on flow cessation. 5. Anomalous Rheological Behavior We already have noted some striking similarities between the nonlinear rheology of this microemulsion sample and that of conventional entangled polymer melts and solutions. However, a peculiar anomaly is related to the transient viscosity on flow inception, and is illustrated particularly well by data collected at 10 °C (Figure 11a). Note first that the overall sequence of observations here is identical with that presented in Figure 2. At sufficiently low rates, the viscosity grows according to the predictions of linear viscoelasticity, whereas higher rates first give rise to stress overshoots, and then an additional slower (phase separation) process at still higher rates. Of interest here is the early stage of the transient viscosity growth. As seen clearly in Figure 11a, the transient viscosity in the nonlinear regime falls above the envelope predicted by linear viscoelasticity. This is contrary to experience in entangled flexible polymers.17 The data in Figure 2 show a similar anomaly, but the slower fluid relaxation time (and hence more leisurely pace of the experiment) at 10 °C renders these transient processes slow enough that the possibility of instrumental artifacts may confidently be discounted. The fact that transient viscosity on flow inception falls below the linear viscoelastic envelope in flexible polymers reflects the fundamentally shear-thinning character of the nonlinear response to shear.31 The most direct way to interrogate a viscoelastic fluid’s nonlinear strain dependence is through step-strain deformation. Figure 11b presents measurements of the nonlinear relaxation modulus of the microemulsion. At strains up to 50%, the step strain data are well described by the linear relaxation modulus, whereas at larger strains negative deviations are observed. Except at rather short times, the data are time-strain factorable, allowing a damping function to be extracted according to,

h(γ) )

G(t,γ) G(t)

(2)

where G(t) is the linear relaxation modulus. The measured damping function is presented in Figure 11c, along with the best-fit predictions of two common empirical expressions used to describe flexible polymers, the Wagner model32 and the Papanastasiou-Macosko-Scriven model.33 Although the onset of nonlinearity is somewhat more abrupt in the microemulsion than these models can (31) Magda, J. J.; Lee, C.-S.; Muller, S. J.; Larson, R. G. Macromolecules 1993, 26, 1696-1706.


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in flexible polymers. Note that typical factorable integral models of nonlinear viscoelasticity would only be capable of predicting the positive deviations from linearity seen in Figure 11a if the damping function exhibited at least some regime of strain-hardening behavior. This is not seen in the step strain data, and it represents an interesting puzzle. Compounding the mystery is the fact that in situ X-ray scattering measurements of the microemulsion structure factor following step strain do indicate strainhardening character in an X-ray analogue to the shear stress.34 The origins of these discrepancies are not known. 6. Comparison with Related Systems The transient and steady responses of shear stress in the bicontinuous microemulsion have remarkable similarities to that of wormlike micelles, as discussed in the preceding sections. The behavior of the normal stress, however, is very intriguing. As mentioned earlier, the steady-state normal stress increases linearly with shear rate in the wormlike micelle,9 whereas a stress plateau characterizes the microemulsion. The transient normal stress exhibits a response similar to the shear stress. This phenomenon has been reported in semidilute polymer solutions,25 whereas an opposite trend has been documented with immiscible polymer blends with a droplet/ matrix morphology.30,35 In the latter case, the shear stress shows an undershoot after the initial overshoot, coincident with the first overshoot in the normal stress. Semidilute polymer solutions25,31 have been found to exhibit a second normal stress overshoot. Flow-induced phase separation occurs in these systems and the second overshoot is coincident with the growth of phase-separated structures. This feature is absent in the microemulsion; although a mild undershoot exists, there is no distinct second overshoot. The shear stress also exhibits a second overshoot in the polymer solution, whereas it is not seen with the microemulsion. The important difference is that in solutions, phase separation generally leads to increased stress, whereas our situation exhibits the opposite trend. Polymer blends sometimes exhibit morphological hysteresis in steady rheology with increasing/decreasing shear rate;29 no such hysteresis is observed in this sample. In short, one can conclude that, although superficial similarities exist with other related soft materials, this system is unique in itself. 7. Conclusion

Figure 11. Anomalous rheological behavior of the polymeric bicontinuous microemulsion. (Top) Viscosity as a function of time for startup of steady shear at 10 °C and 0.01 (0), 0.1 (O), 0.316 (4), 2 (3), 3.16 (]), and 5 s-1 (+). Solid line represents the predictions of the linear viscoelastic spectrum measured using dynamic mechanical spectroscopy.7 The linear viscoelastic limit does not form an envelope of the high-shear rate viscosity data. (Middle) Stress relaxation experimental data at different strains: 20% (O), 50% (4), 100% (3), 150% (]), and 200% (+), along with the prediction of the independently measured linear viscoelastic spectrum (solid line). (Bottom) Damping function as a function of strain at 10 °C (O) and 20 °C (0) with fits for the former using Papanastasiou-Macosko-Scriven (solid line) and Wagner (dashed line) models.

quantitatively represent, the overall nonlinear response of the microemulsion is not that different from that found

The model polymeric bicontinuous microemulsion exhibits very rich rheological properties. The transient behavior on flow inception changes remarkably as a function of shear rate. Together with data for step changes in shear rate within and between different regimes, these data support our hypotheses for the morphologies in the four regimes and provide further insights into the kinetics of structure formation. The transient rheology has analogies with related classes of soft materials, particularly wormlike micelles. Considering the gross differences in the structure of wormlike micelles or “living polymers” and the bicontinuous microemulsion, the striking similarity in rheological behavior between the two is surprising. Both the systems show shear thinning due to (32) Wagner, M. H. Rheol. Acta 1976, 15, 136-142. (33) Papanastasiou, A. C.; Scriven, L. E.; Macosko, C. W. J. Rheol. 1983, 27, 387-410. (34) Caputo, F. E.; Burghardt, W. R.; Krishnan, K.; Bates, F. S.; Lodge, T. P. Phys. Rev. E 2002, 66, 041401/1-18. (35) Takahashi, Y.; Noda, I. Flow-Induced Structure in Polymers; Nakatani, A. I., Dadmun, M. D., Eds.; American Chemical Society: Washington, D. C., 1995; p 140.

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orientation. An isotropic-nematic phase transition occurs in the former, whereas phase separation takes place in the latter. However, the microemulsion shows no signature of entanglement nor the single relaxation time, i.e., Maxwellian response36 of wormlike micelles. In addition, oscillations are observed in the stress at high shear rates in the wormlike micelles, which are absent in the microemulsion. In terms of the phase diagram, the polymeric bicontinuous microemulsion is found across a narrow composition range at a precarious balance point between the phaseseparated regime and the swollen lamellar phase.3 The application of shear flow to this microemulsion has the effect of moving the system into the phase-separated window, i.e., as though reducing the amount of copolymer (36) Cates, M. E. Macromolecules 1987, 20, 2289-2296.

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at equilibrium. The transient rheological behavior of the swollen lamellar phase, at a composition just outside the microemulsion region, is completely different, as will be described in a future publication. These features underscore the fact that polymeric bicontinuous microemulsions can serve as excellent model systems for this class of weakly structured soft materials. Acknowledgment. We would like to thank David Giles for assistance with rheological measurements, Kristoffer Almdal for polymer synthesis, and Franklin Caputo for useful discussions. This work was supported by the MRSEC program of the National Science Foundation under Award DMR-9809364, at the University of Minnesota. LA026081P

Transient Rheology of a Polymeric Bicontinuous Microemulsion

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