This polymeric bicontinuous microemulsion clearly exhibits rich rheological phenomena; four regimes are observed as a function of the shear rate. We associate different morphologies with these regimes, as depicted in Fig. 14; various experimental techniques have been used, each one providing a different perspective on the flow-induced transformations of the structure. Here in Sec. IV we provide justification for the proposed structure in each of these regimes; analogies are also drawn with both block copolymer mesophases and immiscible polymer blends.

Figure 14.

Steady shear rheology was used to establish a Newtonian behavior in regime I. The characteristic time scales of the experiment were much longer than the relaxation time of the microemulsion, resulting in a liquid-like response. This was confirmed by the in situ SANS experiment, where the isotropic diffraction ring resulting from the bicontinuous microemulsion was unaffected by Couette flow. From this we deduced that the morphology was unperturbed (Fig. 14) in this regime.

The linear dynamic mechanical data from the bicontinuous microemulsion are remarkably similar to the rheological response of symmetric lamellar forming block copolymers just above the order–disorder transition (T_ODT), for example, reported by Rosedale and Bates (1990) for poly(ethylene propylene) (PEP)–poly(ethyl ethylene) block copolymers. With block copolymers, terminal viscoelastic behavior (G~² and G~) was obtained at temperatures well into the disordered phase, whereas below T_ODT, the scaling followed G~G~^0.5. Near T_ODT, a lack of superimposability at low frequencies characterized the viscoelastic response, and this has been attributed to fluctuation effects. Because the amplitude of these composition fluctuations is temperature dependent, and because it also influences the level of sustainable stress (i.e., G), fluctuating block copolymer melts become thermorheologically complex. Our data on the bicontinuous microemulsions (Fig. 5) closely resemble the fluctuation dominated disordered state block copolymer response.

This kind of a response can be understood by treating the bicontinuous microemulsion channel as an extension of a "band" of fluctuations around the T_ODT of the lamellar phase. Consider the phase diagram in Fig. 1(b). At the pure block copolymer limit, the system self-assembles into a lamellar phase. The addition of homopolymers swells this phase. The lamellar spacing progressively increases with the added homopolymers, ultimately resulting in phase separation. However phase separation is pre-empted by the fluctuation-induced formation of a bicontinuous microemulsion. This happens when the swelling reaches a point where the interfaces are sufficiently separated that they lose orientational correlation [Matsen (1999)]. Around the order–disorder transition boundary, there exists a band of strong fluctuations (this band leads to suppression of lamellar phase formation as the temperature is decreased through the ODT). We envision the bicontinuous microemulsion channel as an extension of this band of fluctuations; the resemblance of our rheological data with those on lamellar block copolymers supports this notion.

Failure of time–temperature superposition in the microemulsion means that there is some subtlety to the temperature dependence of the structural dynamics and rheology of this sample. Three relaxation phenomena with distinctly different temperature dependences are discernible. The first two, which concern the average relaxation time computed from the linear viscoelastic spectrum and the zero-shear viscosity, are related to the macroscopic response of the microemulsion. The third one, the high frequency viscous response, is related to the relaxation at the molecular level. Connections can be made between the rheology and SANS data to better understand these phenomena.

The average relaxation time of the microemulsion structure [see Eq. (2)] exhibits the strongest temperature dependence. Examination of Table II shows that the temperature dependence of the development of anisotropy in the neutron scattering tracks that of the average relaxation time (shift factors for lead to an apparent activation energy of 170 kJ/mol, while shift factors for the anisotropy index lead to an apparent activation energy of 150 kJ/mol). In order to obtain coupling between the structure and flow, and thus scattering anisotropy, the shear rate must exceed the inverse of the microemulsion relaxation time. These two processes could be characterized by a type of Weisenberg number (bearing in mind that we are dealing with the microemulsion relaxation time; the molecular relaxation times of the polymer constituents are much smaller).

The zero-shear viscosity is also strongly temperature dependent, although somewhat less so than the average relaxation time. It appears that the temperature dependence of the phase-separation index as documented by neutron scattering (activation energy of 115 kJ/mol) is equivalent to that of the microemulsion zero-shear viscosity (activation energy of 120 kJ/mol). This similarity suggests that the phase-separation process is activated by a critical stress level rather than by achieving a critical Weisenberg number (in ordinary polymers these two criteria would be identical; they are different here due to the complexity of this sample). In fact, this idea is more forcefully supported by the connection between the observed phase separation and the stress plateau seen in steady shear, which will be discussed further. Finally, the viscous response at high frequency exhibits a temperature dependence that is intermediate between those of the constituent homopolymers, but closer to PEE, due to the fact that its high viscosity dominates the purely viscous response of the microemulsion (Fig. 6).

Upon entering regime II, anisotropy develops in the morphology (Fig. 14). The domains that are initially oriented perpendicular to the direction of flow tend to be rotated away due to the shear flow. As a result, there is suppression of concentration fluctuations along the direction of flow and the SANS patterns become anisotropic. Complementary SANS experiments (not shown) with the beam directed tangential to the Couette cell (see Fig. 2) confirm these results. The anisotropy leads to a decrease in the degree of percolation of the bicontinuous structures along the direction of the velocity gradient. This gives rise to shear thinning, particularly because there is great contrast between the viscosity of the two homopolymers. It is worth noting that there is only a superficial similarity between the elongated structures in the microemulsion and worm-like surfactant micelles ("living polymers"). Although both exhibit shear thinning due to orientation, the microemulsion shows no signature of entanglement or the single relaxation time, i.e., Maxwellian response [Cates (1987)] of "living" polymers.

As described in Sec. I, the bicontinuous microemulsion is located in a narrow composition range between lamellar and phase-separated states. One might expect, therefore, that shear flow could lead the sample to become either lamellar or phase separated in a fashion analogous to that of flow-induced shifts in the phase boundaries observed for binary polymer blends. Studies on related soft materials (sponge phases and block copolymers, for example) have indicated a transition from an isotropic to a lamellar state, as discussed in Sec. I. In contrast, shear flow of the bicontinuous microemulsion leads to phase separation after reaching a critical shear stress. We hypothesize that the microemulsion ejects two homopolymer-rich phases when the stress exceeds this critical value. Some block copolymer is also expelled and is distributed between the homopolymer-rich phases, with the remainder still lying at the interfaces between domains. We thus expect three-phase coexistence in regime III as illustrated in Fig. 14, the bicontinuous microemulsion and two homopolymer-rich phases. With an increase in shear rate across regime III, the extent of phase separation progressively increases, while maintaining a constant stress. SANS and SALS results show the concurrent development of strong low-q scattering, which indicates a dramatic increase in the structural length scale. They also indicate remnants of scattering intensity in the q range of the bicontinuous microemulsion, thus supporting the argument for three-phase coexistence.

Flow-induced phase separation also can be rationalized in terms of movement across phase boundaries. This is equivalent to moving downward in the isothermal phase diagram [Fig. 1(a)] from the bicontinuous microemulsion region, first into the three-phase triangle, and then into the two-phase region when the shear rates reach regime IV. In terms of the phase diagram along the isopleth [Fig. 1(b)], this corresponds to horizontal movement from the microemulsion channel into the phase-separated regime.

The progressive increase in the degree of phase separation across regime III is documented by optical microscopy. These experiments reveal string-like structures as a result of phase separation, with the number of strings or the amount of homopolymer-rich phases increasing with the shear rate. This is accompanied by a strong growth in turbidity. SALS experiments also support this argument, with the formation of a streak-like pattern at the onset of phase separation and a progressive increase in intensity with the shear rate across regime III.

The shear-induced anisotropy persists for a long time after flow is brought to an abrupt halt. This includes the formation of droplet-like structures by the retraction and breakup of the strings and a general nonrandom distribution of the domains. Based on these observations, we may conclude that the string-like phase in regime III arises from true phase separation and is not due to flow instabilities such as those reported by Fernandez et al. (1995).

The stress level corresponding to the plateau, in addition to being independent of the shear rate, is also nearly independent of the temperature; the reason for this is not clear at present. The onset of the stress plateau also roughly corresponds to the shear rate when the viscosity of the bicontinuous microemulsion becomes equal to that of the homopolymer PEE. Stress plateaus have been reported in several other complex fluids. In the case of worm-like micelles, Spenley et al. (1993) explained the plateau on the basis of a nonmonotonic dependence of stress on the shear rate in a single fluid phase, resulting in shear banding instability; Olmsted and Lu (1997, 1999) and Berret et al. (1994, 1998) interpreted the stress plateau in similar systems in terms of a transition between an isotropic and a nematic phase, with the plateau representing the two-phase region. In our system, the plateau corresponds to breakdown of the microemulsion and represents the transition between the bicontinuous and completely phase-separated states. Unlike other systems, there is no layer or band formation along the direction of the shear gradient; instead there is the development of string-like morphology. The stress plateau is robust and no hysteresis effects are observed. Long time transients are observed in the rheological behavior upon the inception of flow in the stress plateau. They show a remarkable resemblance to worm-like micelles and will be presented in a future publication. In the case of worm-like micelles, the nematic phase has a much lower viscosity than the isotropic phase. Similar rheological properties are observed for the bicontinuous microemulsion when the system undergoes phase separation and ejects low viscosity material.

Well into regime IV, the SANS patterns show complete loss of intensity at the microemulsion q range. We therefore propose that there is complete decomposition of the bicontinuous microemulsion into two homopolymer-rich phases in this regime. The block copolymer assumes a relatively minor role, and either resides in the interfaces or becomes solubilized (perhaps as micelles) in the homopolymer-rich phases (Fig. 14). Hence the sample behaves just like a typical binary polymer blend; the stress increases with the shear rate, albeit accompanied by some shear thinning [Veenstra et al. (1999); Miles and Zurek (1988)]. In SALS experiments a dark streak is seen in the middle of the bright pattern when the shear rates are well inside regime IV. At very high rates, the scattering pattern resembles a butterfly. There also is an overall decrease in the intensity of the patterns at these high rates, probably due to high turbidity and multiple scattering.

Butterfly-like SALS patterns similar to those in our experiments have been reported for polymer blends only in the case of transient evolution of bright streak patterns [Hong et al. (1998, 2000); Chen et al. (1995)]. In our system, this pattern occurs at steady state [the well-known steady state butterfly patterns in semidilute polymer solutions reported by Moses et al. (1994) and by Kume et al. (1997) are qualitatively different]. Hong et al. (1998, 2000) have attributed some of their butterfly patterns to a chevron-like morphology. This structure arises because of the development of secondary flow with a velocity component in the vorticity direction. The superposition of primary and secondary flows results in helical flow. The projection of helical flow on a plane perpendicular to the beam would show chevron-like structures that produce a butterfly-like scattering pattern.

An alternative explanation invokes a change in length scale as a function of the shear rate [Chen et al. (1995)]. One can imagine the butterfly pattern as being a "squashed" spinodal ring, with the distance between the intensity maxima in the two directions being inversely proportional to the respective length scales. In the case of a bright streak pattern, the length scale along the direction of flow is so large that the wings of the butterfly almost touch each other and the dark streak in the middle is not visible. When there is breakage of strings at high rates, and hence a reduction in the length scale along the direction of flow, the dark streak can easily be seen. It is unclear at present whether these explanations based on transient patterns are applicable to the steady state.

Microscopic observations are difficult in regime IV due to the high velocity of the strings and the strong turbidity. Increasing the illumination helped us to discern fine ( < 1 µm) strings even at rates close to 100 s^–1 at 15 °C. Stopping the shear leads to the immediate formation of fine grain-like structures [Fig. 13(b)]. These structures coarsen initially; then the interfaces become hazy, ultimately leading to a homogeneous bicontinuous microemulsion phase. One can call this kind of phenomenon "reverse spinodal decomposition." The mechanism for the appearance of the spinodal-like morphology is not fully understood. Our experiments produce results that resemble the behavior of immiscible polymer blends reported by Kielhorn et al. (2000). A plausible explanation for this strange relaxation behavior is that the strings, which are finer at higher rates, break up into many tiny droplets due to Rayleigh instability, and rapidly coalesce with adjacent droplets. The connection to neighboring droplets results in a spinodal-like morphology. Another possible explanation is that the radius of the droplets that appear upon cessation of the shear after near homogenization by shearing, is smaller than the critical radius for domain growth and as a result they dissolve immediately. A spinodal decomposition-like phenomenon follows this, yielding a fine-grain structure [Kielhorn et al. (2000)]. This kind of relaxation behavior which is analogous to that of immiscible blends, supports our argument of complete phase separation in regime IV. Upon the cessation of shear, the short-time response is that of a binary blend, and the system attempts to form macroscopically phase-separated domains via spinodal decomposition. However the system soon recognizes the presence of the block copolymer, which starts diffusing towards the interfaces to form the bicontinuous microemulsion. Two competing phenomena are thus observed: at short times the phase separation through spinodal decomposition is dominant, whereas over longer periods homogenization into bicontinuous microemulsion triumphs. In terms of free energy, the local minimum is the macroscopically phase-separated state whereas the global minimum is the bicontinuous microemulsion phase.

The transition between regimes III and IV is intriguing. SANS shows remnants of the bicontinuous microemulsion even after there is an upturn in stress. It is not clear whether the transition between the two regimes is the point at which there is complete phase separation, or the rate at which the bicontinuous microemulsion no longer remains continuous phase (and hence no longer controls stress in the sample).

Studies of sponge phases [Yamamoto and Tanaka (1996); Mahjoub et al. (1998)] have indicated a shear-induced transition to lamellar phase. It is unclear why the sponge phase becomes lamellar whereas the bicontinuous microemulsion prefers to phase separate. Oil/water/surfactant bicontinuous microemulsions show shear thinning only at very high shear rates [Anklam et al. (1995); Thevenin et al. (1999)]. This is interpreted in terms of the shrinking of conduits along the direction of the shear gradient and their subsequent pinching off. The shear rates for the onset of shear thinning were of the order of 10³–10⁴ s^–1 in the o/w/s case whereas in our system it was of the order of 0.1 s^–1. This implies that the characteristic relaxation times in the two systems differ by several orders of magnitude. Studies of oil/water/surfactant bicontinuous microemulsions have been hindered by the necessity to use very high shear rates. This problem is overcome by the use of polymers. In addition, the polymeric system presents an opportunity by which to systematically change the "strength" or degree of segregation of the domains of the microemulsion by varying the temperature. In the o/w/s systems, the temperature–composition phase plane, often called a "fish plot" [Gompper and Schick (1994)] has a narrow range of temperatures over which the bicontinuous microemulsion exists. This precludes any attempt to study the effects of temperature on the rheological properties. Hence ternary polymer blends present many advantages over conventional o/w/s systems.