Dynamic frequency sweep experiments were performed on the bicontinuous microemulsion sample at various temperatures. Stress relaxation experiments established a linear viscoelastic response for strains less than 50%. Therefore, frequency sweep data were collected using 10% strain at lower temperatures and 20% strain at higher temperatures, well within the linear regime. Raw linear viscoelastic data obtained at various temperatures are presented in Fig. 4 as the magnitude |^*| and phase angle of the dynamic viscosity versus frequency . The complex viscosity exhibits a plateau at low frequencies followed by relaxation at high frequencies [Fig. 4(a)], behavior typical of a polymeric liquid. Closer inspection reveals, however, that the extent of the relaxation is more pronounced at low temperatures. As a result, these data cannot be reduced to a unique master curve by time–temperature superposition, a point reinforced by the phase angle plot [Fig. 4(b)]. Terminal, liquid-like behavior is evident at the lowest frequencies and highest temperatures ( approaches 90°). A decreasing phase angle indicates an increasing elastic character in the viscoelastic response up to intermediate frequencies, after which the trend reverses towards a more viscous response at high frequencies. Shifts in the curves towards lower frequency at low temperatures reflect a slowing down of the dynamics. In addition, the minimum value of phase angle decreases at lower temperature, indicating that the mixture becomes more elastic at lower subcoolings within the microemulsion channel. Viscoelastic materials that obey time–temperature superposition allow simple shifting of the phase angle along the frequency axis; clearly the microemulsion is not thermorheologically simple.

Figure 4.

Figure 5(a) presents a cross plot of G vs G [Han et al. (1995)]. Data collected at different temperatures should collapse onto a single curve in such a plot for viscoelastic materials that obey time–temperature superposition. Here we see that the data converge at high values of G and G (e.g., at high frequencies), whereas they fail to superimpose at low frequencies. We surmise that the low frequency response reflects the viscoelasticity associated with the microemulsion structure, which is temperature dependent. A decrease in temperature increases the degree of segregation between the constituents, leading to a larger viscoelastic contribution by the supermolecular structure. Conversely, the high frequency response presumably is dominated by more local-scale dynamics, which should be less affected by changes in the thermodynamic state of the microemulsion. These data suggest that the most meaningful way by which to shift the viscoelastic data along the frequency axis is to attempt to achieve superposition at high frequency; this is shown in Fig. 5(b). The viscoelastic response of the bicontinuous microemulsion is quite similar to that observed for undiluted block copolymer melts [Rosedale and Bates (1990); Almdal et al. (1990)] and block copolymer solutions [Jin and Lodge (1997)] just above the order–disorder transition temperature. The reduced frequency plots show terminal linear viscoelastic behavior with G~² and G~ in the limit 0.

Figure 5.

Oscillatory shear measurements on the pure homopolymers over the same frequency and temperature range show no measurable elastic response. Hence, the elasticity revealed in these experiments must be due to the microemulsion structure present in the ternary blend. The data presented in Figs. 4 and 5 can be fit to a generalized Maxwell model using two relaxation modes per decade (Table I). A viscous "solvent" mode (relaxation time=0) must be added to properly represent the data. The main contribution to this mode is from the purely viscous response of the constituent homopolymers. There may also be a small contribution from the high frequency viscoelastic response due to the microemulsion structure (G continues to increase at the highest accessible frequency, indicating fast-relaxing viscoelastic modes in the sample). Assuming, however, that this term is dominated by the rheology of the homopolymer constituents, we would anticipate that it would be close to a weighted average of the homopolymer viscosities. This solvent viscosity is plotted along with the homopolymer viscosities and the zero-shear viscosity of the bicontinuous microemulsion as a function of temperature in Fig. 6. The solvent viscosity that reflects the high frequency dynamics, does, indeed, lie between the homopolymer viscosities. Since PEE is relatively close to its glass transition (T_g–30 °C), it exhibits a much higher viscosity than PDMS (T_g–125 °C), and a stronger temperature dependence. Assuming that both components in the bicontinuous structure experience the same deformation, the simplest estimate for the "composite" viscosity would be

Due to the dramatic contrast in viscosity between the two homopolymers, the purely viscous response of the microemulsion can be associated with a viscosity that is roughly half that of the PEE homopolymer. Similar reasoning suggests that the temperature dependence of _mix should be dominated by the PEE component. Instead, we find that the high frequency microemulsion viscosity is closer to 20%–25% of the PEE, although Fig. 6 clearly shows that the temperature dependence of the high frequency viscosity closely parallels that of the PEE. The lower-than-expected viscosity may reflect incomplete segregation with PDMS acting as a plasticizer, thereby reducing the viscosity of the PEE-rich domain in the microemulsion. These arguments ignore the role of the PEE–PDMS diblock (10% of the total sample), although this should actually promote a higher viscosity because of its higher molecular weight and topological restriction to the interface.

Figure 6.

Since the microemulsion does not follow time–temperature superposition, there is no unique way to characterize the temperature dependence of the sample. The two most meaningful measures would seem to be tracking the temperature dependence of the average relaxation time (using results from Table I) and the temperature dependence of the zero-shear viscosity (data from Fig. 6).

Given a discrete relaxation spectrum fit to the linear viscoelastic data, it is possible to compute an average relaxation time according to

where _i are the relaxation times and _i the corresponding viscosities. Table II summarizes the temperature dependence of several quantities of interest, represented by shift factors relative to a reference temperature of 15 °C. Comparing the shift factors of the high frequency solvent viscosity to the PEE homopolymer viscosity confirms that they share a similar temperature dependence. Conversely, both the zero-shear viscosity of the microemulsion and its average relaxation time exhibit much stronger temperature dependences than either of the homopolymers. The ordinary temperature dependent dynamics of the constituent polymers are compounded by changes in the thermodynamic state of the system. As the temperature decreases, the degree of segregation between PEE and PDMS in the microemulsion increases, resulting in an increasingly well-defined (and elastic) interfacial structure, and greater barriers for diffusion except through the tortuous bicontinuous morphology. The strong temperature dependence of the average viscoelastic relaxation time is particularly noteworthy: the relaxation time changes by two orders of magnitude over a span of just 20°.

Pätzold and Dawson (1996) presented a Landau–Ginzburg analysis of the dynamic linear viscoelasticity associated with small amplitude sinusoidal deformation of a bicontinuous microemulsion. The predicted behavior is, in gross terms, similar to the Rouse model: terminal relaxation at low frequencies, and G=G~^1/2 (=45°) at high frequencies. In a real system, this would be accompanied by a viscous contribution from the pure components, which would enhance G relative to G by a constant amount, and lead to an upturn in phase angle at high frequencies. Qualitatively, these predictions agree well with the behavior seen in Figs. 4 and 5; in particular, the high frequency G data come close to exhibiting power law scaling with an exponent 1/2.

Steady shear rheological data also were collected at different temperatures and shear rates. Figure 7(a) shows the viscosity of the bicontinuous microemulsion as a function of shear rate at various temperatures. The corresponding stresses are shown in Fig. 7(b). These data are presented without any attempt at time–temperature shifting. A cursory comparison between Figs. 4 and 7 shows that the Cox–Merz rule [Cox and Merz (1958)] fails for the microemulsion. As will be shown below, this reflects radical steady shear-induced changes in the fluid structure at high rates. Four regimes can be identified as a function of the shear rate. Figure 8 depicts these four regimes at 15 °C. The viscosities of the homopolymers are also plotted for comparison.

Figure 7.

Figure 8.

The bicontinuous microemulsion shows Newtonian behavior at low shear rates, which is designated as regime I. Note that the zero-shear viscosity strongly decreases with the temperature [see Fig. 7(a)] due to the temperature dependence of the morphology of the bicontinuous microemulsion, as discussed earlier. Newtonian behavior persists up to the onset of shear thinning, which signals the beginning of regime II. Experiments conducted at other temperatures revealed that the critical shear rate for shear thinning increases with an increase in temperature (e.g., 0.03 s^–1 at 10 °C; 0.1 s^–1 at 15 °C), and also that the shear rate window for regime II narrows with an increase in temperature.

Regime III is characterized by a shear rate independent shear stress (i.e., the viscosity power law index –1), and a marked enhancement in shear thinning. This dramatic change suggests a significant transformation in the morphology. Remarkably, the data collected at different temperatures almost overlap within regime III, indicating a stress level that is nearly independent of both the shear rate and the temperature. Increasing the shear rate further leads to an abrupt upturn in stress. We refer to this as regime IV. The rate dependence of in this regime is much less than in regime III.

Significant normal stresses are also observed, particularly in regime IV. These are found to track the behavior of the shear stress in regime IV and in most of regime III. Even though the homopolymers do not show much elasticity, the presence of interfaces deformed by the shear flow leads to noticeable normal forces. However at lower shear rates the normal force falls below resolution of the instrument.

SANS experiments were conducted with an in situ Couette flow shear cell operated at different temperatures and shear rates. Figure 9 shows representative scattering patterns obtained at 15 °C. In the quiescent state [Fig. 9(a)], a circular ring of scattering intensity is evident, characteristic of the bicontinuous microemulsion. The associated structure factor obeys the form [Teubner and Strey (1987)]

where q=(4/)sin(/2) is the scattering wave vector, is the scattering angle, and I(q) is the azimuthally averaged one-dimensional scattered intensity. Two characteristic length scales can be obtained from the fitting parameters: a domain periodicity d and a correlation length scale that reflects the short-range order among the domains; at 25 °C, d=74 nm and =32 nm. In addition, an amphiphilicity factor f_a can be defined using the fitting parameters [f_a=c₁/(4a₂c₂)^1/2] to classify the structure of the fluid. This number, which quantifies the degree of segregation between the domains [Schubert et al. (1994)] is negative and increases in magnitude as the microemulsion becomes more segregated at lower temperatures [see, for example, Morkved et al. (1999, 2001) for a fuller treatment of these thermodynamic properties].

Figure 9.

At low shear rates, within regime I defined from rheology (Fig. 8), the SANS patterns are unperturbed by steady shear (i.e., they remain isotropic). Higher shear rates (regime II) produce anisotropy in the SANS patterns [Fig. 9(b)], with a loss of intensity along the flow direction. However, the q values associated with the intensity maxima in regime II are almost the same as in regime I, indicating that the concentration fluctuations with wave vectors parallel to the flow direction are suppressed by the shear flow, but that the structure is unperturbed along the vorticity direction. In regime III, intense low-q scattering develops around the beamstop, which increases with the shear rate [Figs. 9(c) and 9(d)]. Simultaneously the microemulsion scattering diminishes, first along the direction of flow, and then in all directions. Development of low-q scattering suggests the emergence of a morphology with a much larger length scale. This is indicative of flow-induced phase separation from a single bicontinuous microemulsion phase to a micron-scale multiphase state. This transition appears to be continuous, suggesting that the volume fraction of the microemulsion phase progressively decreases across regime III.

At very high shear rates, in regime IV, the microemulsion scattering completely disappears and only the strong low q scattering remains [Fig. 9(f)]. This low-q intensity grows to almost 10 times the intensity recorded for the unperturbed mixture. There is saturation in the scattering intensity at low q at these high rates. In fact, at 15 °C the low q intensity passes through a maximum at 100 s^–1, then decreases somewhat at still higher rates. An overall behavior pattern similar to that in Fig. 9 is also seen at other temperatures.

In order to quantify the SANS anisotropy in regime II, we define an "anisotropy index." Figure 9(a) shows two small boxes labeled "A" and "B" within which average intensities I are computed. We define the anisotropy index as 2I_B/(I_A+I_B). The value of the index is unity for an isotropic pattern. Figure 10(a) depicts the anisotropy index as a function of the shear rate at various temperatures. There is a strong decrease in the index as the sample enters regime II.

Figure 10.

A "phase-separation index" can be defined in a similar way. Figure 9(a) also shows a small box labeled "C," which represents the detector pixels used in defining this index as I_C()/I_C(=0). The index has a value of unity before the onset of phase separation. Figure 10(b) shows the phase-separation index plotted as a function of the shear rate. A marked increase in this index occurs in regimes III and IV.

Figure 10 is quite informative. First, a comparison of Figs. 10(a) and 10(b) shows that the temperature dependence of the development of anisotropy is stronger than the temperature dependence of the development of excess low-q scattering intensity. This may be quantified by exploiting the fact that the shapes of the curves in Figs. 10(a) and 10(b) are similar at all temperatures. Figure 10(c) presents a "master plot" obtained by shifting the data from Figs. 10(a) and 10(b). The shifting was carried out by superimposing the data at high shear rates using 15 °C as a reference temperature. Note that separate sets of shift factors were used to shift the anisotropy and phase separation data. The quality of the superposition is quite satisfying. Both sets of shift factors are listed in Table II. Several of the features referred to earlier are apparent. The transition from regime I to II is evident in the anisotropy index. The onset of phase separation, which we associate with the beginning of regime III, is signaled by the upturn in the phase-separation index. The saturation in low-q scattering at high shear rates is also evident. There is a slight mismatch between the critical shear rates for these transitions predicted by SANS and by rheology, the critical shear rates from the former being systematically less than those from the latter. This implies that neutron scattering is more sensitive to changes in structure than rheology. But the development of two different phenomena can clearly be seen from each of these two separate experiments.

Small angle light scattering provides access to much smaller q values, hence larger length scales than SANS. Light scattering experiments were carried out at various temperatures and shear rates using the procedures described in Sec. II. Figure 11 shows representative data at 15 °C. In the quiescent state, no significant light scattering was recorded, a consequence of the characteristic dimensions of the bicontinuous microemulsion. There was only a small amount of stray background scattering due to reflection from the optical elements. This lack of scattering persisted as the shear rate was increased through regimes I and II, consistent with the SANS analysis. Upon entering regime III(1.5 s^–1), a streaked light scattering pattern emerged around the beamstop perpendicular to the direction of flow, which increased in intensity with an increase in the shear rate across regime III. The appearance of the bright streak coincides with the onset of the stress plateau in rheology (Fig. 8), and this critical shear rate increases with the temperature. Similar streak-like patterns have been reported in sheared polymer solutions [Kume et al. (1997)] and in blends [Hong et al. (1998); Kielhorn et al. (2000); Fernandez et al. (1995); Kim et al. (1997)] and attributed to the presence of a string-like morphology aligned along the direction of flow. Accurate quantitative analysis of the scattering intensity cannot be obtained with the present experimental setup. However the q range of the scattered intensity at these shear rates suggests a length scale on the order of microns, consistent with the hypothesis that flow-induced phase separation occurs in regime III, with phase-separated domains oriented along the direction of flow. Further increases in the shear rate lead to loss in the overall intensity of the scattering images. This is due, at least in part, to increased turbidity of the sample. At shear rates well into regime IV( > 20 s^–1), a dark streak is superimposed along the center line of the bright pattern. We return to this feature in Sec. IV. The dark streak separating the two lobes of scattered intensity becomes wider with an increase in shear rate, resulting in a butterfly-like pattern. As the shear rate increases, the wings of the butterfly move apart and ultimately disappear at the highest rates. In addition, the overall intensity of the patterns decreases with an increase in temperature. We attribute this last effect to a reduction in contrast, due to more mixing of the components.

Figure 11.

Light scattering data were also collected during relaxation after shear. In regime III, cessation of steady shear led to a gradual retraction of the streak-like pattern into the beamstop that occurred over a period of 15–30 min, depending on the shear rate. In contrast, recovery from very high shear rates (~200 s^–1) in regime IV led to the immediate (1–2 s) development of isotropic scattering around the beamstop. The isotropic pattern gradually diminished in intensity and finally disappeared in about 20–30 min.

The bicontinuous microemulsion sample appears homogeneous and almost transparent to the naked eye, but has a faint blue tint due to scattering of light from the nanostructure. Rapid stirring makes the sample turbid, and the turbidity increases with the stirring rate. This, again, is indicative of flow-induced phase separation. More systematic measurements were conducted by operating the shear stage under an optical microscope. Figure 12 illustrates the steady state morphologies obtained at 15 °C at various shear rates.

Figure 12.

In the quiescent state, the sample is homogeneous (except for a few dust particles). Steady shear flow does not produce any significant change for shear rates corresponding to regimes I and II. However in regime III (shear rates 3, 10, and 20 s^–1), the turbidity of the sample increases, accompanied by the development of a string-like morphology aligned along the direction of flow. There is a drastic transformation from the nanometer-size microemulsion to this micron-size string-like phase. The turbidity further increases with the shear rate, as evidenced by a progressive darkening of the images. There also is an increase in the length scale along the direction of flow across this regime, as seen from the images in Fig. 12. Further increases in the shear rate in regime IV result in strong turbidity, and the images become very dark. Fine string-like structures still could be seen even at shear rates close to 100 s^–1 when the illumination was increased. The sample thickness was reduced in order to obtain better images, since the velocities would be lower for a given shear rate there by overcoming limitations of the video recording system. However below a certain gap width spurious wall effects became evident.

Recovery of the microemulsion also was recorded after the cessation of shear flow. As noticed for the light scattering, the modes of recovery were very different at intermediate and high shear rates. Figure 13 illustrates the recovery after steady shearing at 7.5 and 150 s^–1. In the former case (regime III), the anisotropy in the morphology persists for a long time, and there is also formation of droplet-like structures due to retraction and breakup of the string-like domains. These structures gradually dissolve over a period of about 20–30 min. On the other hand, after cessation of shearing at 150 s^–1 (regime IV), there was an immediate increase in the overall brightness of the image, followed by the formation of a nearly isotropic grain-like structure, slightly elongated along the direction of flow. These structures resemble the spinodal-type phase-separated domains that are seen in binary polymer blends after cessation of shear at high rates [Kielhorn et al. (2000)]. These structures appear within seconds, whereas the droplets in regime III take a few minutes to develop. The grain-like structures coarsen, and simultaneously the interfaces become hazier. The sample reaches the original homogeneous state in about 45 min.

Figure 13.

Turbidity measurements were performed using the light scattering setup by measuring the transmitted intensity (i.e., without the beamstop) using a photodiode. The photodiode gives a voltage signal that is proportional to the intensity. A nearly constant intensity was recorded in regimes I and II, and a sharp drop was observed in regime III, clearly due to phase separation. The intensity transmitted progressively decreased with an increase in shear rate across regime III, ultimately reaching a steady value. This increase in turbidity confirms our speculation regarding the loss of brightness found in the microscopy images in Fig. 12. The absolute magnitude of the scattered intensity in regime III, of course, strongly decreases with an increase in sample thickness.