Time-Periodic Structures and Instabilities in Shear-Thickening

Apr 19, 2002 - being enhanced by shear stress and disappearing at low stresses or above the theta point. At the higher stresses, macroscopic rings are...
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Time-Periodic Structures and Instabilities in Shear-Thickening Polymer Solutions Loic Hilliou† and Dimitris Vlassopoulos* Foundation for Research and Technology - Hellas (FORTH), Institute of Electronic Structure & Laser, P.O. Box 1527, Heraklion 71110, Crete, Greece

We revisit the problem of flow-induced structural changes in semidilute polystyrene/dioctyl phthalate (PS/DOP) solutions below their theta temperature by focusing on the high-shearstress regime where thickening occurs. We observe a strong coupling of flow instabilities to the induced structure resulting from the enhanced concentration fluctuations. This behavior is manifested as a time-periodic signal of both the optical (birefringence and dichroism) and mechanical responses and represents an identifying signature of the strong-thickening region, being enhanced by shear stress and disappearing at low stresses or above the theta point. At the higher stresses, macroscopic rings are formed throughout the sample and sustained over the duration of the shearing. The emerging picture suggests that flow-enhanced fluctuations and eventually phase separation are the cause of shear thickening. I. Introduction The subject of shear-induced structure formation in complex fluids such as polymer and surfactant solutions has received a great deal of attention in recent years because of its implications in processing, in the fundamental understanding of the coupling of flow and concentration fluctuations, and in the role of instabilities.1-31 An experimental system that has been studied nearly exhaustively is the semidilute solution of polystyrene in dioctyl phthalate, hereafter abbreviated as PS/ DOP.2-8,13-17,20 This particular solution serves as a model system because DOP is a theta solvent for linear PS at 22 °C. Using a wide range of techniques, namely, rheology, rheo-optics, light scattering, and optical microscopy, systematic investigations of the effects of shear on this solution have revealed the existence of three different regimes:3 (i) homogeneous solution for γ˘ < γ˘ c (the subscript c denotes the onset of shear thinning); (ii) shear-induced concentration fluctuations for γ˘ c < γ˘ < γ˘ a, yielding the formation of domains, accompanied by butterfly patterns in light scattering; and (iii) an “anomalous” behavior for γ˘ > γ˘ a, characterized by rheological and light-scattering anomalies. In the second regime, the amplitude of concentration fluctuations is enhanced along the flow direction, whereas the domain structure is extended perpendicular to the flow. The third regime, which covers the range of shear rates from below the onset of shear thickening to the highest value reached, is characterized by a change in the sign of the dichroism and increased values of both birefringence and dichroism, attributed to the formation of stringlike structures oriented along the flow direction.13 The region of shear thickening has not been studied in much detail, partly because of the complicated nature of the optical and mechanical signals. In this paper, we focus on the systematic in situ study of the shear-thickening regime with the aim of clarifying * To whom correspondence should be addressed. Tel.: +3081-0-391469. Fax: +30-81-0-391305. E-mail: dvlasso@ iesl.forth.gr. † Present address: Max-Planck Institut fu¨r Polymerforschung, P.O. Box 3148, D-55021 Mainz, Germany.

the onset of this time-periodic behavior, establishing its phenomenology, and contributing to the identification of its origin. We show that this regime is characterized by a time-periodicity in both the optical and mechanical signals, which are out-of-phase and exhibit a decreasing period as the stress increases. Well into the thickening region, periodic macroscopic rings are formed that are sustained over the course of shearing and disappear upon flow cessation. These observations support the scenario of flow-enhanced concentration fluctuations (and eventually phase separation) being the cause of shear thickening. II. Experimental Section Materials. A high-molecular-weight (Mw ) 2 × 106 g/mol), nearly monodisperse (Mw/Mn ) 1.15) polystyrene (PS) sample was obtained from Polymer Source, Canada. It was dissolved in dioctyl phthalate (DOP) at a concentration of 8 wt %, which corresponds to an effective entanglement molecular weight of about 450 000 g/mol, based on the solvent-mediated dilution of entanglements.32 The solution was prepared by using dichloroethane as cosolvent and subsequently removing it gradually at room temperature under a vacuum. Methods. In situ rheo-optical measurements were carried out by combining a Rheometric Scientific constant-stress rheometer (DSR-200) with an optical train.33,34 The latter consisted of a He-Ne laser (5 mW), a polarizer, a photoelastic modulator (at 50 kHz), a quarter-wave plate, a flow cell, a circular polarizer (CP), and a photodiode. The flow cell consisted of two parallel quartz plates of diameter 38.1 mm, with the sample loaded between them (the sample thickness was about 1 mm). When the CP is in the optical path, this configuration effectively measures the birefringence (∆n′); on the other hand, when the CP is removed, the measurement directly provides the dichroism (∆n′′). Measurement details are found in refs 3, 33, and 34. For the parallel-plate geometry, the orientation angle relative to the flow direction, θ′ ) χ′ - 45° (χ′ being the orientation angle relative to the polarizer direction), is 0° or 90° for symmetry reasons. The retardation δ′ relates to the birefringence ∆n′ in the same plane

10.1021/ie0110078 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/19/2002

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Figure 1. Shear-rate-dependent viscosity of the PS/DOP solution investigated for the three temperatures (0) 10, (O) 15, and (4) 20 °C. Lower Inset: Dynamic frequency sweeps at 15 °C, showing the storage (G′, O) and loss (G′′, b) moduli; solid lines have (terminal) slopes 1 (for G′′) and 2 (for G′), and their intersection determines the inverse relaxation time. Upper Inset: Shear viscosity versus shear stress at 15 °C for two different sample thicknesses: (O) 1 and (4) 1.8 mm.

through δ′ ) 2π∆n′d/λ, with d being the optical path length (sample thickness) and λ the laser wavelength. When the CP is removed from the optical path, the dichroism is measured, and δ′′ is the extinction of the light, related to the dichroism through δ′′ ) 2π∆n′′d/λ. The orientation angle of the dichroism relative to the flow is θ′′ ) χ′′ - 45°. Rheological measurements included primarily step and steady stress and rate sweep tests in the range 100-1800 Pa, with a varying sample thickness as discussed below; unless otherwise noted, the measurements were performed with a thickness of 1 mm. Note that the reported shear rates refer to the rim values in this parallel-plate setup; the actual values at the point of optical measurement are lower by a factor of 0.84. Temperatures below the theta point were investigated, namely, 10, 15, and 20 °C, using the original DSR recirculating fluids bath, appropriately modified to fit in the optical train.34 The rheo-optical data reported here refer to 10 and 15 °C. III. Results and Discussion Rheological Response. Figure 1 depicts the rheograms for the three temperatures investigated, namely, 10, 15, and 20 °C. Shear thickening is unambiguously observed in all cases. It is clear that, whereas the zeroshear viscosity η0 increases as the temperature is lowered, at the same time, the critical shear rate γ˘ th corresponding to the onset of thickening behavior shifts to lower values; this is in agreement with earlier observations on the same system.5,6 The lower inset of Figure 1 depicts a dynamic frequency sweep at 15 °C from which the longest relaxation time of PS chains is extracted, λ ≈ 0.6 s, in agreement with the birefringence relaxation time (Figure 2 below) and the onset of shear thinning γ˘ c-1. For the experimental conditions relating to Figure 1, the Reynolds number is Re ) FRγ˘ d/η ≈ 10-5 (F is the fluid density, R is the plate radius, and d is the sample thickness), whereas the critical Weissenberg

number for the onset of shear thickening can be estimated as With ) λγ˘ th ≈ 36 at 15 °C, which virtually matches the theoretical prediction for an elastic instability in a Taylor-Couette device.1 Note that γ˘ th > γ˘ a, where the latter defines the onset of anomalies in optical signals.3 The upper inset of Figure 1 shows the shear viscosity as a function of shear stress at 15 °C for two different sample thicknesses, namely, 1 and 1.8 mm. It is apparent that, whereas the two rheograms are virtually indistinguishable at low stress levels, in the shear-thickening region, the measured viscosities exhibit a strong gap dependence. In particular, the viscosity minimum shifts to higher stresses with increasing sample gap. This finding is qualitatively consistent with the prediction of gap dependence of the critical Weissenberg number in a Taylor-Couette geometry1 or the formation of a transient network from shear-induced elongated polymer-rich domains at different gaps.14 In this parallel-plate geometry, it is not possible to make a quantitative assessment of these predictions, although the idea of shear-induced phase separation is favored, as discussed below. Shear-Thinning Region. In the two regions of relatively low shear rates corresponding to the Newtonian plateau and the early shear-thinning behavior, the flow-induced anisotropy, as detected through the intrinsic birefringence, exhibits the features already established in the literature.3,5,6 When in the Newtonian region, the PS chains orient along the flow direction, exhibiting a weak negative birefringence with a transient that is characterized by the absence of overshoot and a steady state that is reached quickly. In the thinning regime, before the viscosity minimum is reached, a small overshoot characterizes the transient birefringence response at early times (Figure 2a). In this shear rate range, corresponding to Wi > 1, a weak positive dichroism signal evolves without overshoot (Figure 2b), indicative of enhanced concentration fluctuations orienting perpendicular to the flow (θ′′ ) -45°),

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Figure 2. Transient optical response of PS/DOP in various shear rate regimes at 15 °C: (a) birefringence in the early shear-thinning region (shearing at 10 s-1 for 20 s); (b) corresponding dichroism signal; (c) dichroism in the strong-thinning region, approaching the viscosity minimum (shearing at 40 s-1 for 25 s); (d) corresponding dichroism signal just at the onset of thickening (shearing at 50 s-1 for 25 s).

with the constituent chains orienting along the flow, as in the Newtonian regime (see also Scheme 1, below).3,5 With further increasing of the shear rate, still before the minimum viscosity is reached, the birefringence overshoot is enhanced. At the same time, the positive dichroism signal is complex, characterized by a large overshoot, followed by a virtual zeroing of the signal (θ′′ ) 0° or -180°), and a second overshoot upon flow cessation preceding the relaxation of dichroism (Figure 2c). The dichroism overshoot might result from the competition between shear enhanced concentration fluctuations and the polymer stress.8,9,16,18 These phenomena are enhanced at the critical shear rate γ˘ th corresponding to the viscosity minimum at the onset of thickening (Figure 2d). A change in sign of the dichroism is observed at the critical shear rate γ˘ th. Actually, after the initial overshoot, the transient dichroism signal goes to zero and then changes sign. The orientation angle first goes to 90° (θ′′ ) 135° or -45°) and is expected to progressively reach 0° at even higher shear rates. The shear stress transients show clearly the initial overshoot, but the second one is barely observed. Note that shear-enhanced concentration fluctuations are manifested as a butterfly pattern in light scattering,3,8,9,16,20 as already mentioned in the Introduction. The fact that the dichroism changes sign, passing through zero, relates to the increasing importance of the convection of the concentration fluctuations.9 We can therefore identify the onset of shear thickening with the change in sign of the linear conservative dichroism (see also Figure 7 below). This finding seems to contrast the results of Kume et al.,3 who reported negative dichroism at γ˘ < γ˘ th. However, we have verified our results at three different temperatures, all below the theta point, and further, we provide below an explanation of the occurrence of negative dichroism at the onset of shear thickening.

Shear-Thickening Region. Creep Analysis. We now focus on the shear-thickening regime. Here, we start with stress-controlled measurements (creep), which allow the viscosity to be defined uniquely at each shear stress value. The shear-stress-dependent viscosity at 15 °C is depicted in Figure 3a for a gap of 1.8 mm. Judging from this rheogram, the transient birefringence behavior in different stress regions can be analyzed, namely, the viscosity minimum (Figure 3b), an intermediate stress in the thickening regime (Figure 3c), and the highest stress reached (Figure 3d). In the first case (Figure 3b), the shear rate increases with time and reaches its steady-state value quickly and smoothly. At the same time, the induced birefringence goes through an overshoot and quickly and smoothly reaches its steady-state value; this behavior is typical of that observed throughout the shear-thinning regime. At this point, it should be emphasized that the change in sample thickness does not affect the qualitative features of the observed optical and mechanical responses. However, in the case of Figure 3c, well into the thickening region, an oscillatory behavior is observed. Actually, the shear rate oscillates with nearly timeperiodic oscillations having a period of about 17 s; at the same time, the birefringence response is also characterized by a time-periodic signal with approximately the same period. This oscillatory behavior is attributed to an apparently unstable flow in this regime, which is the reason that a true steady state in the shear rate is not reached (self-sustained oscillations of rate and birefringence in time). Further, concerning the higher stress level attained (Figure 3d), the same periodic phenomena are observed, but with a reduced period of about 10 s and a slightly enhanced birefringence amplitude; it is evident that the period of the oscillating signals is a function of the shear stress. Judging from this evidence, it can be safely concluded

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Figure 3. Birefringence and shear rate response in the shear-thickening region at 15 °C for a PS/DOP sample thickness of 1.8 mm: (a) stress-dependent viscosity; (b) transient behavior at the critical stress σth ) 1200 Pa corresponding to the viscosity minimum; (c) transient behavior at an intermediate stress in the thickening regime, σ ) 1600 Pa; (d) transients at the maximum stress level reached, σ ) 1800 Pa.

that oscillations in both the optical and mechanical signals appear whenever Wi > With, where With assumes a value of 60 in this case, which is about twice the value found with a smaller gap (see Figure 1). This gap dependence is opposite to that predicted (Wic ) 35) by theories describing pure elastic instability in TaylorCouette flows or parallel-plate flows.1 On the other hand, values of the critical Weissenberg number higher than the predicted value for elastic instabilities have been reported in the literature and attributed to strong thinning and the vanishingly small ratio of normal stress differences N2/N1, which together tend to stabilize the flow, thus shifting the critical Wi to higher values.1 In the shear rate region of relevance here, the ratio N2/ N1 tends to zero;6 thus, a value of Wi > Wic is expected. The fact that this is not exactly the case for the smaller gap of 1 mm (as discussed above) points again to the strong gap dependence of With, which, in turn, effectively rules out the scenario of elastic or inertial instabilities (also because of the vanishingly small value of Re).1,26,27 Note also that the shear rate and birefringence oscillations in Figure 3c,d are out-of-phase, as a maximum in the former corresponds to a minimum in the latter. The overall response of the birefringence signal is shown in Figure 4, where the (average) steady-state values of the shear viscosity and birefringence were obtained at 10 °C. Note that the steplike increase of the negative birefringence in the shear-thinning regime is indicative of the measurement time per shear stress data point, which was selected to ensure steady-state measurements (typically, it was 60 s). The thickening regime is characterized by an oscillatory behavior of the birefringence, as already discussed above; nevertheless, one can recognize the sharp increase in birefringence in the thickening region, as well as its nearly constant (but fluctuating) stress-independent value at the onset of thickening. The former observation is qualitatively consistent with earlier experiments.3 Finally, upon flow

cessation (after the last viscosity data point in Figure 4), the birefringence reverses sign instantly to become positive and then relaxes slowly to zero (see also Figure 3). Such a behavior has been detected but not discussed in PS/DOP systems in the past5 and has also been reported in other systems, e.g., wormlike chains.35,36 It suggests that, when the flow stops, the PS chains instantly relax their orientation, whereas positive birefringence originating from form contributions relaxes more slowly as polymer-rich domains disappear. Stress-Optical Rule. Using the approach of Chu et al.,34,37 we can estimate the stress optical coefficient (SOC) in the parallel-plate geometry employed. This assumes the coincidence of the principal directions of the stress and refractive index ellipsoids, implying that the measurements are taken in the linear regime. In such a case, the following equation holds

nij ) Cσij and θ ) θ′

(1)

where θ is the angle between the flow direction and the principal stress axis in the flow plane; θ′ is the extinction angle resolved in the flow plane, i.e., the angle between the flow direction and the principal axis of the refractive index ellipsoid; and C is the stress optical coefficient (SOC). For simple shear flow between two parallel plates, this expression can be written as

σ12 ) ∆n′12 sin 2θ/2C

(2)

N1 ) ∆n′12 cos 2θ/C

(3)

N2 ) ∆n′23/C ) (∆n′13 - ∆n′12 cos 2θ)/C

(4)

where 1, 2, and 3 refer to the flow, velocity gradient, and vorticity directions, respectively, and N1 and N2 are the first and second normal stress differences, respectively. Under the assumption that ∆n′23 is vanishingly

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Figure 4. Steady-state rheo-optical response at 10 °C: viscosity-stress rheogram and corresponding stress birefringence measured in situ (see text). The vertical arrows with the STOP FLOW notation are added to indicate the subsequent fast relaxation of the optical signal.

small, we obtain

∆n′13 ) ∆n′12 cos 2θ

(5)

In fact, ∆n′13 is the quantity measured during an experiment with the parallel-plate geometry. On the other hand, the same experiment probes the ‘12’ component of the stress tensor, σ12 ) ηγ˘ . A combination of eqs 2 and 5 yields

∆n′13 ) 2Cσ12 cot 2θ

(6)

The trigonometric term in the above equation can be expressed in terms of the shear rate. Using the expected scaling N1 ≈ γ˘ 2 for concentrated solutions,38 from eqs 2 and 3, we obtain cot 2θ ≈ γ˘ . Therefore, the measured flow birefringence scales with shear rate according to the following equation

∆n′13 ) 2Cη0λhγ˘ 2

(7)

where λh is the longest characteristic relaxation time, typically associated with the onset of shear thinning. Figure 5 illustrates the extraction of the SOC using this approach, by plotting the induced birefringence in the ‘13’ velocity-vorticity plane as a function of the square of the shear rate. The inset of this figure shows the lowrate regime where essentially eq 7 holds; from the slope and the longest relaxation time from Figure 1, we estimate C ≈ 5 × 10-11 cm2/dyn. This value is smaller than that reported in the literature using Couette geometry and a different solvent (2.7 × 10-10 cm2/dyn),39 but given these considerations, as well as the assumptions used in our approach and the related uncertainty of the procedure followed, it is considered reasonable. Note also that, beyond about 10 s-1, the stress optical rule no longer holds; one explanation is the fact that N2 is no longer zero and that large shear rates affect C.39,40

Linear Conservative Dichroism. We now examine the flow-induced dichroism signal in more detail. A positive value of ∆n′′ develops in the Newtonian (albeit weak) and shear-thinning regimes (Figure 2b,c). However, as the thickening regime is reached, the ∆n′′ signal is reversed (Figure 2d). In analogy to the birefringence analysis in Figure 3, the creep experiments indicate that, with increasing stress, the shear rate and birefringence and dichroism signals become progressively time-periodic. From smooth variations in the viscosity minimum region, they exhibit periodic patterns at the highest stress value studied. A typical example is depicted in Figure 6 (for a gap of 1.8 mm), which corresponds to a shear stress of 1800 Pa. The timeperiodic response reflects the behavior of the shear rate, much like the birefringence experiment (Figure 3), and signifies the apparent flow instabilities and their coupling to the induced structures. The typical period of the ∆n′′ and shear rate out-of-phase signals for this figure is about 12 s, similar to that for ∆n′ (Figure 3d). The oscillations of both optical signals, as well as the (out-of-phase) mechanical signal, correlate with visual observations of the shear solution, which indicate the formation of turbid rings having a nearly constant spatial distance of about 2 mm (see Scheme 1d below). This observation brings analogies to oscillatory patterns in other complex fluids, such as the coupling of shearinduced structures and instabilities in surfactant solutions.10,11 In that case, the phenomenon was explained by invoking a phenomenological model describing the interaction of flow and concentration changes via a coupling parameter involving the derivatives of chemical potential with respect to shear rate and stress with respect to concentration.41 In the present situation, this theory is not directly relevant. However, if one assumes that increasing temperature at constant concentration has a qualitatively similar effect to decreasing concentration at constant temperature, one can attempt a

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Figure 5. Steady stress birefringence as function of the square of the shear rate at 15 °C. Inset: Enlargement of the low-rate region for the extraction of the stress optical coefficient.

Figure 6. Transient behavior of dichroism and shear rate response at the maximum stress level (1800 Pa) in the shearthickening region at 15 °C for a sample thickness 1.8 mm.

qualitative comparison with the data. Indeed, the calculated derivative of the stress with respect to the shear rate increases upon cooling, a trend observed in the experiments as well. Despite these differences, however, the striking qualitative similarities in the response of conventional and living (surfactants) polymers might be suggestive of a common microscopic origin of the coupling between instabilities and shearinduced structures (phase-separating and aggregating large-scale domains, respectively). It should be noted that a more quantitative analysis of the data would require a truly homogeneous shear field, which can be achieved with a cone-and-plate geometry; however, a high-optical-quality glass cone was not available. Nevertheless, the qualitative nature of the reported results is not affected. The cumulative plot of Figure 7 (at 10 °C) demonstrates the interplay of flow and structure development. Of particular interest is the decrease of ∆n′′ as the viscosity minimum is approached, which goes through

zero at the minimum and then reverses its sign at the critical stress σth, and the enhancement of the (periodic) fluctuations in the ∆n′′ values at the highest stress values. On the other hand, when the flow stops (after the last viscosity data point in Figure 7), ∆n′′ instantly reverses its sign and further exhibits a long relaxation from positive values to zero. Along with the corresponding birefringence results of Figure 4, these findings suggest the immediate relaxation of the string structure (rapid loss in the collective orientation of domains) upon flow cessation, followed by the slow relaxation of concentration fluctuations.9 Shear-Thickening Region. Stress Transients. To examine the stress overshoots in the thinning and thickening regions, which relate to structural changes of the sheared solution,9,20 we present in Figures 8 and 9 the mechanical and optical transients during strainrate-controlled experiments at 15 °C with a gap of 1.8 mm. Starting from Figure 8, in the early shear-thinning region (γ˘ ) 10 s-1), a stress overshoot is probed, corresponding to chain stretching, at about 1 s upon inception of shear flow, i.e., roughly 10 strain units. Note that, whereas overshoots are predicted at about 2 strain units, experimental evidence suggests that they occur at larger number of strain units, provided that γ˘ λR > 1, where λR is the Rouse time.38,40 The birefringence signal also exhibits an overshoot, but at longer times. This is because shear stress and birefringence are probed in different planes in this geometry,33 and the latter relates to N1, which is expected to exhibit a delayed overshoot with respect to the shear stress with a smaller amplitude, as documented in the literature38,40 and confirmed in this work as well (see Figure 10 below). As the shear rate is increased (100 and 120 s-1) and the thickening regime is entered, both the stress and birefringence transient signals exhibit damped oscillations toward steady state, which are characterized by two main peaks; these can be thought of as two overshoots.4,20 In

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Figure 7. Steady-state rheo-optical response at 10 °C: viscosity-stress rheogram and corresponding stress-induced dichroism measured in situ (see text). The vertical arrows with the STOP FLOW notation are added to indicate the subsequent fast relaxation of the optical signal.

fact, the birefringence signal is nearly the mirror image of the stress; therefore, the second weak overshoot can be associated with chain stretching, originating from shear-induced elongated domains of concentration fluctuations9 and eventually yielding phase separation.3,4,20 It is linked to the observed growing structures induced by flow (see also Figure 9 and Scheme 1 below).3,20 This overshoot is predicted9 to occur at γ˘ λD > 1, which conforms to the experiments as the “cooperative” time λD is on the order of 0.1 s.42,43 At the highest shear rate of 140 s-1, steady-state values have not been reached after 15 s of shearing, and the increase of the stress (nearly overload) and its long-time oscillations are reminiscent of instabilities.4 Figure 9 depicts the dichroism transients corresponding to the conditions of Figure 8, along with the corresponding birefringence for comparison. As already mentioned, the latter relates to the transients of the first normal stress difference. Whereas at 10 s-1 a smooth increase in both optical signals is observed (note that, despite the noise in the dichroism signal, no overshoot is observed at this shear rate), some interesting features are observed at higher levels of shear. The second overshoot in birefringence relates to a growth in dichroism; on the other hand, the first overshoot in birefringence is associated with the onset of dichroism. Further, the dichroism transients are slower, as they probe larger structures (fluctuating domains) than polymer chains (birefringence). At the highest shear rate, 140 s-1, an apparent flow instability develops, as evidenced by the oscillations in both optical signals at long times. Figure 10 shows the shear and first normal stress transients for a rate of 1 s-1, just at the onset of shear thinning, indicating that the overshoot in the latter occurs at later times. Note that, for this particular situation, the Cox-Merz rule is valid. The inset of Figure 10 shows the N1 relaxation upon flow cessation, which is fitted to an exponential decay N1 ) N1,se-(t/τ) (where the subscript s refers to the steady-state value), yielding a relaxation time τ of about 0.8 s, in agreement with the birefringence relaxation (Figure 2a). Note that

a second overshoot in N1 is not observed,20 but this is not unexpected as these data were taken long before the thickening region. These measurements were carried out in a strain instrument (Rheometric Scientific ARES); an oblique-angle measurement with the optical setup44 was not attempted because of the relatively low signal-to-noise ratio. Assessment of the Overall Rheo-Optical Response. Direct visual observations are also indicative of the complex response of PS/DOP in the strongthickening region. The sample is turbid as expected, as dichroism is detected and increases in value. With increased stress, the migration of the fluctuations in different radial positions reveals the formation of rings of typical size less than 1 mm and spacing of about 2 mm (Scheme 1d), which are sustained over the time of observation. The sample becomes transparent after the stress is removed, which is expected from the fast relaxation of the optical signals and is consistent with the literature.3,10 A compilation of the above phenomena is presented in Scheme 1, which attempts to rationalize the observed flow-structure interplay. Whereas parts a and b are well-documented in the literature, the main interest in this work relates to parts c and especially d, in the thickening regime, which is dominated by the coupling of the inhomogeneous (due to flow oscillations at high stresses) velocity field and the enhanced induced fluctuations. These temporal (and also spatial) oscillatory patterns are reminiscent of rings observed in wormlike surfactants;10,11,45 however, in contrast to that situation, the observed dependence of the onset of thickening on the sample thickness (increase of With with increasing sample thickness, in contrast to expectations, as already discussed) should rule out the possibility of inertial and elastic instabilities in favor of phase separation in a nonuniform shear field.9,14,26-31 In addition, these rings do not relate to string formation in shear polymer blends at small sample thickness.46 In that case, an induction time was needed for the strings, which were observed over of a range of viscosity ratios from 0.1 to 10 (much smaller than the PS/DOP

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Figure 8. Shear stress and birefringence transients of PS/DOP (15 °C and sample thickness of 1.8 mm) during start of shear at different rates: (a) thinning, 10 s-1; (b) early thickening, 100 s-1; (c) strong thickening, 140 s-1.

system) and were sensitive to interfacial forces. In the present situation, shearing promotes stress oscillations, and the PS-rich domains (rings) formed increase in size and eventually have a lateral dimension on the order of the size of the sample thickness and give rise to the macroscopic shear thickening.14 This is reflected in the higher critical stress σth for the onset of thickening for larger sample thicknesses. Furthermore, the data reported here rule out the possibility of shear banding instabilities (Figures 1, 4, and 7)26,45,47 on the grounds of the stress-shear rate behavior (no true stress plateau) and the time periodicity (and space periodicity, for the rings) of the optical and mechanical signals. It seems, therefore, that shear-induced phase separation in the most general sense (microstructural changes yielding clusters) is the generic underlying cause of shear thickening, as evidenced in a variety of soft complex fluids, where a variety of forces (enthalpic,

Figure 9. Transient dichroism signals, along with their birefringence counterparts, corresponding to the conditions of Figure 8.

lubrication, electrostatic) are coupled to the flow. In addition to the polymer solutions discussed here, experimental results on micellar solutions,10,26,27,48 ionomers,49-51 and colloidal suspensions52-55 strongly support this conclusion. IV. Conclusions The investigation of semidilute PS/DOP solutions sheared below their theta temperature at high stress levels in the thickening region revealed a strong coupling of the flow instabilities and the induced structure resulting from the enhanced concentrations fluctuations, eventually yielding phase separation. This behavior was manifested in primarily two ways, which represent the new findings of this work: (i) time-periodic signals of the optical and mechanical response and (ii) the formation of macroscopic rings around the plates, covering the entire sample. In addition, the onset of shear thickening

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Figure 10. Stress σ (O) and first normal stress coefficient N1 (solid curve) transients of PS/DOP at 15 °C during start of simple shear flow at a rate of 1 s-1 (onset of thinning). The dotted vertical line indicates the stress overshoot, which precedes the N1 overshoot (see text). Inset: Relaxation of N1 upon cessation of shear flow.

Scheme 1. Cartoon Representation of the Flow-Induced Structure (Dichroism) and Chain Orientation (Birefringence) at Different Levels of Imposed Shear Stressa

chanical signals were out-of-phase and had a period that decreased as the stress increased. Their amplitude was enhanced with shear stress, and they disappeared at stresses below the onset of shear thickening or above the theta point. The strong coupling of the stress and the induced structural changes in the thickening region was also confirmed by the stress and optical transients, which exhibited sharp primary overshoots and subsequent decaying irregular oscillations, without reaching steady state in the time scale of the experiment. The rings were sustained over the time of shearing and had a rough spatial periodicity, and they extended over the whole sample thickness. Whereas they did not relate to elastic or shear banding instabilities, these rings were analogous to those observed with wormlike surfactants and attributed to shear-enhanced fluctuations and phase separation. The latter provided insight into the cause of the shear thickening. Acknowledgment

(a) σc < σ < σa, early shear-thinning region (σc corresponds to the onset of shear thinning), corresponding to the formation of domains under shear with a random orientation of constituent PS chains; (b) σa < σ < σth, strong shear-thinning region before the viscosity minimum, where the PS chains inside the domains orient in the flow direction (horizontal arrow); (c) σth < σ < σosc, onset of shear thickening with the domains ordering into large-scale stringlike patterns in the flow direction; and (d) σ > σosc, strongthickening region with time-periodic mechanical and optical responses, yielding the formation of macroscopic rings over the entire sample (the round arrow illustrates the flow direction). The signs (+) and (-) indicate the positive and negative directions of the optical signals, respectively. a

was identified by the change in sign of the dichroism and the second overshoot in the stress and birefringence transients. The periodicity detected in both the birefringence (segmental orientation) and the dichroism (domain orientation), as well as the shear rate responses, represents an identifying signature of the “strong-thickening” region (σosc). The optical and me-

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Received for review December 13, 2001 Revised manuscript received February 13, 2002 Accepted February 15, 2002 IE0110078