Onion Transition with Varying Temperature under

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Re-entrant Lamellar/Onion Transition with Varying Temperature under Shear Flow Daijiro Sato,† Kahoru Obara,‡ Youhei Kawabata,† Makio Iwahashi,‡ and Tadashi Kato*,† †

Department of Chemistry, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan Department of Chemistry, Kitasato University, 1-15-1 Kitasato, Minami-ku, Sagamihara 252-0329, Japan

‡

S Supporting Information *

ABSTRACT: We have found for the first time the reentrant lamellar/onion (lamellar−onion−lamellar) transition with varying temperature under constant shear rate by using simultaneous measurements of shear stress and small-angle X-ray scattering (Rheo-SAXS) for a nonionic surfactant (C14E5)/water system, which exhibits the lamellar phase in a wide temperature range from 15−75 °C. The onion state exists in a closed region in the temperature−concentration diagram at a constant shear rate. Temperature dependence of the lamellar repeat distance (d) at rest has also been measured at several concentrations. It is shown that the increase of d with increasing temperature is necessary for the existence of the lower transition. We have investigated the change in the lamellar orientation in the lamellar-to-onion and onion-to-lamellar transition processes near the upper and lower transition temperatures. For all four kinds of transition processes, the following change in the lamellar orientation is observed; lamellar state (oriented to the velocity gradient direction) ↔ further enhancement of the orientation to the velocity gradient direction ↔ enhancement of the orientation to the neutral direction ↔ onion state.

1. INTRODUCTION The shear-induced lamellar-to-onion transition may be one of the most striking phenomena among the effects of shear flow on the structure of surfactant lyotropic phases. This transition has been found two decades ago by Roux and co-workers.1−4 They have shown that the shear flow induces transformation of the lamellar structure into the multilamellar vesicles (onions) with a polyhedral shape, which fill all the space without excess water. They have also shown that the radius of onions (R) can be controlled by the shear rate (γ̇) because it follows the power law, R ∝ γ̇−1/2. In their subsequent studies for other systems, long-range-ordered plane structures have been revealed where onions are hexagonally shaped and close packed on triangular lattices.5−7 Following their pioneering studies, the onion formation has been reported for many surfactant systems.8−46 Diat et al.2 have made the dynamic phase diagram, where the range of the shear rate for the onion formation is plotted against the volume fraction of membranes. According to it, the onion can be formed above a critical shear rate, which increases with increasing the membrane volume fraction. For aqueous solutions of ionic surfactants, another important variable is the salinity. For example, in the AOT [bis(2-ethylhexyl) sulphosccuinate]/brine system, Leon et al.17 have found that the viscosity of the sample is suddenly increased after a certain delay time, tg, under constant shear rate and tg increases rapidly with a slight increase in the NaCl concentration. At low salinities, the viscosity of the sample is already rather high compared to the usual lamellar phase and the polarized microscope observation indicates the existence of many onionlike defects. Spontaneous onion formation in the AOT/ © XXXX American Chemical Society

brine system with low salinity has also been reported by van der Linden and Buytenhek.47 These results are explained in terms of salinity dependence of the elastic moduli 2κ + κ,̅ where κ and κ̅ are the bending and saddle splay moduli of bilayers, respectively.17,47 However, there have been only a few systematic studies where surfactant concentration and/or salinity are varied, which may be due to the difficulty of changing the salinity continuously. On the other hand, it has been known that binary systems of water and nonionic surfactant of the polyoxyethylene type CnH2n+1(OC2H4)mOH, abbreviated as CnEm, exhibit a variety of phase behaviors just by changing the temperature even without any additives.48 This results from the fact that the spontaneous curvature of surfactant film decreases with increasing temperature.49,50 About 10 year ago, Olsson and co-workers have reported that the lamellae are transformed into onions with decreasing temperature under constant shear rate for the C10E321 and C12E418,22 systems. They explain these results based on the fact that the spontaneous curvature of surfactant films increases with decreasing temperature, which leads to decrease of 2κ + κ̅ (see later). On the other hand, we have recently reported for the first time the lamellar-to-onion transition with increasing temperature under constant shear rate in a C16E7/water system by using simultaneous measurements of small-angle light scattering/shear stress (Rheo-SALS) and small-angle X-ray scattering/ Received: October 21, 2012 Revised: December 6, 2012

A

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pattern in the flow-neutral (vorticity) plane is obtained in the radial configuration, while that in the velocity gradient-neutral plane is obtained in the tangential configuration. For the lamellar structure, there are three principal orientations, perpendicular (or A), transverse (or B), and parallel (or C) orientations, with the layer normal along the neutral, flow, and velocity gradient directions, respectively. These three orientations can be detected by using both radial and tangential configurations.

shear stress (Rheo-SAXS).38,40 This transition cannot be explained by the change of 2κ + κ̅ alone because the onion formation occurs when the temperature is increased. Thus, the conditions for onion formation have not yet been clarified, in spite of many reports on the onion formation in both ionic and nonionic surfactant systems. In the C10E3 system, Oliviero et al.27 have reported the temperature-shear rate diagram at 40 wt % of the surfactant based on the viscosity measurements and the previous SANS and SALS study.21 According to it, the lamellar-to-onion transition occurs near the upper limit of the Lα phase (i.e., the boundary between Lα and L3). In the C16E7 system, on the other hand, the transition occurs near the lower boundary of the swollen Lα phase. These results suggest that there should be a system which exhibits both types of transitions. Here, we report for the first time the reentrant lamellar/ onion transition (i.e., lamellar-onion-lamellar transition with varying temperature under constant shear rate in a C14E5/water system). As described later, this system exhibits more general features than the other CnEm systems and thus is considered to be very useful for clarifying conditions and the mechanism of the onion formation. We have also investigated the change in the lamellar orientation in the lamellar-to-onion and onion-tolamellar transition processes near the lower and upper transition temperatures.

3. RESULTS Phase Behaviors at Rest. Figure 1 shows the partial phase diagram of the C14E5−water system at rest. It can be seen from

2. EXPERIMENTAL SECTION Materials and Sample Preparation. C14E5 was purchased from Nikko Chemicals, Inc. in crystalline form (>98%) and used without further purification. Deuterium oxide purchased from ISOTEC, Inc. (99.9%) was used after being degassed by the bubbling of nitrogen to avoid oxidation of the ethylene oxide group of surfactants (we used D2O as a solvent instead of H2O, following our previous study on the C16E7 system51 where 2HNMR is used to determine the phase diagram). Samples containing a desired amount of surfactant and water (about 8 g) were sealed in an Erlenmeyer flask. For homogenization, we annealed samples at room temperature, where onions cannot be formed even under shear (see later), with occasional shaking for about one week. In the Rheo-SAXS experiments, we first sheared the sample with a shear rate of 1 s−1 for 10 min at 25 °C. Then, the shear was stopped, and the sample was heated to the initial temperature in the lamellar phase without shear. Phase Behaviors at Rest. Phase behaviors at rest were determined by visual inspection and a polarizing microscope (Olympus BX51) with a Metller FP82HT hot stage. To confirm the temperature range of the lamellar phase, SAXS was measured at rest by using a Rigaku NANO-Viewer. Rheo-SAXS. The apparatus for Rheo-SAXS experiments is the same one as reported previously.38 We used a Cuvette cell made of polycarbonate consisting of two concentric cylinders whose diameters are 27 mm and 29 mm. The outer cylinder is fixed, and the inner cylinder is attached to an AR550 rheometer (TA Instruments). To prevent sample evaporation, a vapor seal is incorporated in the cell. The most experiments were made on the BL 15A and 6A at the photon factory (PF) of the High Energy Accelerator Research Organization (KEK), Tsukuba. The scattered X-rays were detected using the CCD area detector (Hamamatsu C7300 with a 9 in. Image Intensifier). The exposure time was 30 s. The approximate q range is from 0.3 to 3 nm−1. One of temperature-scan experiments was performed on the BL40B2 at Spring-8, Sayo. The imaging plate (Rigaku R-AXIS VII) was used as a detector. The approximate q range is from 0.2 to 3.7 nm−1. The exposure time was 3 s. Two scattering configurations were used; one is the so-called radial configuration where the X-ray beam is directed through the center of the cell (along the velocity gradient direction), and the other is the tangential configuration where the beam is directed through the end of the cell (along the flow direction). The two-dimensional (2D) SAXS

Figure 1. Paritial phase diagram of the C14E5/D2O system at rest. L1, micellar phase; Lα, lamellar phase; S, hydrated solid phase; 2ϕ, twophase coexistence. The ○, △, and crosses in the Lα phase indicate the onion state, transition region, and lamellar state, respectively, at the shear rate of 3 s−1. The dotted lines are present as guides. The block arrows indicate the temperature range where transition processes have been investigated in detail.

the figure that the Lα phase exists over a very wide temperature range in this system. Above about 35 °C, the Lα phase extends to the lower concentration as is observed for many CnEm systems. For the sample containing 35 wt % and 40 wt % C14E5, we have found an isotropic phase at around 72 and 76 °C, respectively, in the two-phase coexistence region, which suggests the existence of the L3 phase. Because the phase behaviors were determined every 5 wt %, the phase boundary (solid lines in Figure 1) includes errors ± 2.5 wt %. Temperature Dependences of Shear Stress and SAXS Intensity. First, rheo-SAXS experiments have been performed at 50 wt % as a function of temperature from 25 to 75 °C at a constant shear rate of 3 s−1. This shear rate was chosen because B

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raised by 1 K every 15 min (the average heating rate is about 0.067 K/min) near the lower and upper transition temperature and more rapidly for other temperature ranges. So the shear stress does not always become constant at each temperature (see Figure 2 of Supporting Information). On the other hand, we have also measured the shear stress at constant temperature and shear rate. The results are shown in Figure 3 of the Supporting Information. In Figure 2c, the observed values of the shear stress after 65 000 strain units (about 6 h) at several temperatures are plotted. The agreement between these two sets of data is good. The 2D SAXS patterns were reduced to one-dimensional (1D) patterns in the direction of μ = 0°, 90°, 180°, and 270°, by integrating the scattering intensity over a segment of width Δμ = ± 10° where μ is the azimuthal angle (the neutral direction is set to 90° and 270° both in the radial and tangential configurations). Figure 3 shows typical results (a and b) at rest

the temperature−concentration diagram of the C16E7/D2O system had been made at this shear rate in our previous study. Before the temperature-scan experiments, we measured shear stress as a function of shear rate from 0.05 to 500 s−1 at 50 wt % and 50 °C. The results are shown in Figure 1 of the Supporting Information. It has been shown that as the shear rate increases above 0.1 s−1, the viscosity rapidly increases and takes a maximum at about 3 s−1. Figure 2 (panels a−c) shows

Figure 2. Evolution of 2D SAXS patterns for the (a) radial and (b) tangential configurations, (c) the shear stress and (d) the intensities of the Bragg peaks with increasing temperature at a constant shear rate of 3 s−1. The ○ in the panel (c) indicate the shear stress after 65 000 strain units (about 6 h) at constant temperature. The numbers in the panel (a) and (b) indicate the scattering vector, q, in nm−1. The ●, ■, and ▲ in the panel (d) indicate the intensities for the neutral, flow, and velocity gradient directions, respectively. The orientation of lamellae for each direction is schematically shown in the panel (e).

Figure 3. Evolution of 1D SAXS patterns with increasing temperature (indicated in the figure) for (a and c) the neutral- and (b and d) velocity-gradient directions (a and b) at rest and (c and d) at 3 s−1. The ordinate is shifted in order to separate each pattern.

evolution of 2D SAXS patterns for the (a) radial and (b) tangential configurations and the shear stress with increasing temperature at a constant shear rate of 3 s−1. The pattern for the tangential configuration is not symmetric to the neutral direction (i.e., the intensity on the left-hand side is greater than that on the right-hand side). This is due to the path differences of the scattered X-ray. The radial pattern at 35 °C indicates that the intensity in the neutral direction is much larger than that for the flow direction, while the tangential pattern indicates that the intensity in the velocity gradient direction is much larger than the neutral direction. So the most lamellae are oriented to the velocitygradient direction. When the temperature exceeds 35 °C, the shear stress increases abruptly and takes a maximum at about 42 °C. At this temperature, both radial and tangential patterns become isotropic. Above 42 °C, the shear stress slightly decreases with increasing temperature, whereas the SAXS patterns are still isotropic up to 70 °C. As the temperature exceeds 70 °C, the shear stress suddenly decreases and the SAXS patterns are back to the anisotropic ones, indicating oriented lamellae. In these experiments, the temperature is

and (c and d) at the shear rate of 3 s−1 for the neutral direction (average of the intensities for μ = 90° and 270°) in (a and c) the radial configuration and those for the velocity gradient direction (μ = 180°) in (b and d) the tangential configuration. It can be seen from these figures that only first and second order diffractions are observed, indicating that the system is still in the Lα phase under shear and that transition into other phases does not occur. In Figure 2d, the intensities of the first-diffraction peaks for the three directions are plotted against the temperature in the logarithmic scale. Below about 35 °C, the peak intensities for the neutral and flow directions are only about 1% and 0.1% of the intensity for the velocity-gradient direction, respectively. So the most lamellae are oriented to the velocity-gradient direction (C orientation). In the intermediate temperature range (42−70 °C), the peak intensities for all the directions are almost the same, corresponding to the onion state. Above 70 °C, the C

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intensities for the three directions again become ordered velocity gradient > neutral > flow. In our previous study on the C16E7 system,38 we have performed rheo-SALS experiments in addition to rheo-SAXS experiments. As the temperature increases above a specific temperature, the shear stress increases abruptly and a four-lobe SALS pattern appears, which is characteristic of the onion state. At the same time, the 2D SAXS pattern is changed from anisotropic to isotropic. It has also been reported that the lamellar-to-onion transition accompanies substantial increase in the shear stress. Although we have not performed rheo-SALS experiments in the C14E5 system, the results in Figure 2 strongly suggest the reentrant lamellar/onion transition (i.e., lamellar-onion-lamellar transition) with varying temperature at a constant shear rate. In other words, this system has two kinds of transitions; one is the lamellar-to-onion transition with increasing temperature (35−42 °C), and the other is the lamellar-to-onion transition with decreasing temperature (70− 74 °C). Hereafter, we refer to the former as “the lower transition” and the latter as “the upper transition”. In the C16E7 system, we defined two specific temperatures T* and T** based on the temperature dependence of the shear stress.40 Above T*, the shear stress begins to increase and takes a maximum at T**. Compared to the SAXS pattern, it has been found that the lamellar and onion states exist below T* and above T**, respectively. Following the dynamic phase diagram on the C10E3 system reported by Oliviero et al.,27 we referred to the region between T* and T** as “the transition region”. Because the present system has both the lower and upper transitions, we define four specific temperatures, TL*, TL**, TU*, and TU** as follows: T < TL*: lower lamellar state; TL* < T < TL**: lower transition region; TL** < T < TU**: onion state; TU**< T < TU*: upper transition region; TU*< T: upper lamellar state. In the 1D SAXS patterns at 24.8 °C in the radial direction (Figure 3, panels a and c), a small hump is observed at the slightly lower angle than the first-order diffraction peak. Such a hump has been reported for other CnEm systems such as C16E652 and C16E751,53 and attributed to the diffraction from holes (water-filled defects), which are randomly distributed in bilayers. The SAXS patterns in the tangential configuration (Figure 3, panels b and d) do not give the hump, which confirms the existence of the water-filled defects because the most lamellae are oriented to the velocity gradient direction at this temperature. It should be noted that the SAXS patterns in Figure 3 (panels a and b) have been obtained by using the same apparatus for the rheo-SAXS experiments for the sample presheared at 25 °C. The hump disappears at higher temperatures, which is again the same as that observed in the C16E652 and C16E751 systems. Temperature−Concentration Diagram at a Constant Shear Rate. Figure 4 shows temperature dependences of the shear stress and SAXS peak intensities for the three directions at 55 wt %. In the lower and higher temperature range, the most lamellae are oriented to the velocity-gradient direction. In the intermediate temperature range, on the other hand, the peak intensities for the three directions become almost the same. These results are the same as those at 50 wt % and again suggest the reentrant lamellar/onion transition. At 55 wt %, however, the temperature range for the onion state (difference between TL** and TU**) is much narrower than that at 50 wt %. Moreover, the decrease in the shear stress in the higher transition region is not so steep as that at 50 wt %.

Figure 4. Evolution of (a) the shear stress and the intensities of the Bragg peak with increasing temperature at a constant shear rate of 3 s−1. The ○ in the panel (a) is the shear stress after 65 000 strain units (about 6 h) at constant temperature and shear rate. The ●, ■, and ▲ in the panel (b) indicate the intensities for the neutral, flow, and velocity gradient directions, respectively.

Time evolution of the shear stress is shown in Figure 4 of the Supporting Information. Measurements of the shear stress at constant temperature and shear rate have been made for the 55 wt % sample as well (see Figure 5 of the Supporting Information). The ○ in Figure 4a indicate the shear stress after 65 000 strain unit. Temperature-scan experiments of the shear stress have been performed at 47 and 58 wt % as well. The results are shown in Figure 5, together with those at 50 and 55 wt %. It can be seen

Figure 5. Temperature dependences of the shear stress at a constant shear rate of 3 s−1 at 47 wt % (○), 50 wt % (▲), 55 wt % (■), and 58 wt % (▽).

from the figure that the shear stress in the lower temperature range (the lower lamellar state) increases with increasing concentration as expected. At 47 wt %, the temperature dependence of the shear stress is very similar to that at 50 wt %, but the temperature range of the onion state is so wide that we could not find TU* and TU** within the single Lα phase (see the phase diagram at rest in Figure 1). At 58 wt %, on the other hand, the increment of the shear stress is much smaller than D

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that for other concentrations, suggesting that there is no “complete” onion state. On the basis of these results, the temperature and concentration for the lamellar sate, the onion state, and the transition region are shown in the phase diagram at rest in Figure 1. As can be seen from the figure, the temperature range for the onion state becomes rapidly narrow with increasing concentration. On the other hand, there is a lower concentration boundary in the Lα phase. Therefore, the onion state exists only in a closed region in the temperature−concentration diagram at a constant shear rate. Transition Processes. Because the C14E5 system has both lower and upper transitions, there are four kinds of transition processes at a constant shear rate: (i) lamellar-to-onion transition with increasing temperature in the lower transition; (ii) onion-to-lamellar transition with decreasing temperature in the lower transition; (iii) onion-to-lamellar transition with increasing temperature in the upper transition; and (iv) lamellar-to-onion transition with decreasing temperature in the upper transition. First, we have investigated the processes (i) and (ii) by using rheo-SAXS at 50 wt % (see the lower block arrow in Figure 1), with slower heating or cooling rates compared to the experiments shown in Figures 2 and 3. Figure 6 (panels a and b) show temperature dependences of the shear stress and SAXS peak intensities for three directions in the process (i). The closed symbols indicate the results obtained by a stepwise increase of the temperature by 0.5 K every 15 min (the average heating rate is about 0.03 K/min). The time evolutions of the shear stress and SAXS peak intensities are shown in Figure 6 and 7 of the Supporting Information. As described before, the most lamellae are oriented to the velocity gradient direction in the lower temperature range. At 35 °C, the intensity in the velocity gradient direction increases suddenly and takes a sharp maximum at 35.6 °C, just before the increase in shear stress. As the temperature increases by 0.5 K, the intensity in the gradient direction rapidly decreases and instead, the intensity in the neutral direction increases. As the temperature increases further, the intensities for the neutral and velocity gradient directions decrease, whereas the intensity in the flow direction increases, corresponding to the formation of onions. To confirm the existence of the maximum of the peak intensity for the velocity gradient direction at 35.6 °C, we have also performed rheo-SAXS experiments with a much slower heating rate (the average rate is about 0.005 K/min). The time evolution of the shear stress, temperature, and the peak intensities for the three directions are shown as a function of the shear strain in Figure 8 of the Supporting Information. The last data at each temperature step are plotted as open symbols in Figure 6 (panels a and b). The agreement between the two data sets with different heating rates is fairly good, indicating that the system almost reaches steady state even in the experiments indicated by closed symbols. The results of the cooling process (i.e., the onion-to-lamellar transition with decreasing temperature are shown in Figure 6 (panels c and d). As the temperature decreases, first the neutral peak takes a maximum and then the velocity-gradient peak does. Although a small hysteresis is observed, the temperature dependences of the shear stress and the peak intensities in the heating and cooling processes are very similar. In other words, the orientation of lamellae changes reversibly with varying temperature.

Figure 6. (a and c) Evolution of the shear stress and (b and d) the intensities of the Bragg peak for the (a and b) heating and (c and d) cooling processes near the lower transition (see Figure 1) at a constant shear rate of 3 s−1. The ●, ■, and ▲ in (b and d) indicate the intensities for the neutral, flow, and velocity gradient directions, respectively. The ○, □, and △ in panels (a and b) are obtained with a much slower heating rate (see Figure 8 of the Supporting Information).

Next, we have investigated the processes (iii) and (iv). At 50 wt %, however, the temperature range for the upper transition region within the single Lα phase is quite narrow. Therefore, the experiments have been made at 55 wt % (see the upper block arrow in Figure 1). The results are shown in Figure 7 (Figure 7, panles a and b, are just magnifications of the higher temperature range of Figure 4, panels a and b, respectively). The time evolution of the shear stress is shown in Figure 9 of the Supporting Information. E

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temperature is still the same as that in the heating process (i.e., the neutral peak takes a maximum at the lower temperature than the velocity-gradient peak). For the neutral and flow directions, these two sets of data are in good agreement. Comparing Figure 7 with Figure 6, one sees that the hysteresis in the upper transition is much larger than that in the lower transition. This may be due to the fact that the upper transition region at 55 wt % is much wider than the lower transition region as can be seen from Figure 1. However, these figures indicate that the following sequence of the change in the orientation of lamellae is observed both for the lamellar-toonion and onion-to-lamellar transition and both in the upper and in the lower transition; lamellar state (oriented to the velocity gradient direction) ↔ further enhancement of the orientation to the velocity gradient direction ↔ enhancement of the neutral direction ↔ onion state.

4. DISCUSSION Conditions for Onion Formation. As described in the introduction, the C10E3/water system exhibits the lamellar-toonion transition with decreasing temperature. In accordance with the temperature−shear rate diagram at 40 wt % of C10E3 reported by Oliviero et al.,27 the transition temperature rapidly increases with increasing shear rate from 1 to 10 s−1 and then levels off at least until 100 s−1. The transition temperature at higher shear rates is around 40 °C, which is near the upper limit of the Lα phase (i.e., the boundary between Lα and L3 phases). On the other hand, in the C16E7/water system where the lamellar-to-onion transition with increasing temperature occurs, T** decreases from 76 to 70 °C with an increasing shear rate from 0.3 to 3 s−1, and then levels off. T* takes a shallow minimum but does not change very much (66−67 °C) in the shear rate range between 0.3 and 30 s−1. Above about 3 s−1, T* and T** are close to the lower boundary of the swollen Lα phase. As can be seen from Figure 1, the phase behaviors of the C14E5 system at rest in the higher temperature range are similar to those in the C10E3 system (see Figure 1 of ref 21), and those in the lower temperature range are similar to those in the C16E7 system.38,51 These results suggest that the lamellar-to-onion transition with decreasing temperature in the C10E3 system corresponds to the upper transition, whereas the lamellar-toonion transition with increasing temperature in the C16E7 system corresponds to the lower transition in the C14E5 system. Therefore, the C14E5 system has more general features than other CnEm systems and so is considered to be useful to investigate the conditions for the onion formation as well as the transition mechanisms. The origin of the lamellar-to-onion transition with decreasing temperature in the C10E3 system has been discussed by Le et al.,18,21,22 in terms of the elastic properties of bilayers. In general, the free energy of a unilamellar vesicle relative to a flat bilayer per unit area is expressed as54

Figure 7. (a and c) Evolution of the shear stress and (b and d) the intensities of the Bragg peak for the (a and b) heating and (c and d) cooling processes near the upper transition (see the block arrow in Figure 1) at a constant shear rate of 3 s−1. The circles, squares, and triangles in panels (b and d) indicate the intensities for the neutral, flow, and velocity gradient directions, respectively. The open and closed symbols in panels (c and d) indicate the experiments performed on the BL 40B2 at Spring-8 and on the BL 6A at the Photon Factory, respectively.

As can be seen from Figure 7 (panels a and b), the intensities both for the neutral direction and the velocity gradient direction take maxima and that the former takes a maximum at the lower temperature than the latter. It should be noted that in the cooling process (panels c and d), the intensities for the neutral direction obtained at different runs take maxima at different temperatures. This may be a result from the fact that the initial temperature for one of these runs is not in the lamellar state but in the transition region. However, the sequence of the change in orientation of lamellae with varying

F = 4π (2κ + κ ̅ )

(1)

where κ and κ̅ are the bending modulus and saddle splay modulus of a bilayer, respectively. By using the bending modules κm and the saddle splay modulus κm̅ of a monolayer, κ and κ̅ can be expressed as54−56 κ = 2κ m (2a) κ ̅ = 2κ m̅ − 2δH0mκ m F

(2b)

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where δ is the thickness of a bilayer and H0m is the spontaneous curvature of a monolayer. In the CnEm systems, H0m increases with decreasing temperature,49 which leads to decrease of κ̅ and thus a decrease in the free energy of vesicles if the temperature dependence of the bending modulus is much smaller than that of the saddle splay modulus. However, such a theory cannot explain the lower transition (i.e., lamellar-to-onion transition with increasing temperature) even qualitatively. In our previous study on the C16E7 system, it has been shown that the increase in the repeat distance at rest with increasing temperature is important for the lamellar-to-onion transition with increasing temperature.38 So, we have performed SAXS measurements at rest as a function of temperature at different concentrations. The results are shown in Figure 8. This figure

analyses for the onions formed by shearing AOT/brine/ glycerol and Sochamine 35/water/glycerol systems. In accordance with their TEM images, the shape of onions is not spherical but polyhedral, suggesting that the deformation energy is concentrated near certain planes of stacked edges. Panizza et al.57 have derived an equation which relates the plateau modulus and the radius of onions, taking into account the polyhedral shape of onions. During the derivation, they have shown that deformation energy per volume is given by σ F(R ) ∝ eff (5) R where R is the radius of onions and σeff is the effective surface tension defined as58 σeff ≡

KB̅

(6)

where B̅ is the compression modulus. In the derivation of eq 5, they assume that the width of the wall is equal to the penetration length, λ, defined as58

λ≡

K B̅

(7)

and that the local curvature of the edge is given by 1/λ, although λ is canceled out in the expression of the total energy. For the membranes with Helfrich’s undulation interactions, B̅ can be expressed as59 B̅ =

Figure 8. Temperature dependences of the lamellar repeat distance at rest at 47 wt % (○), 50 wt % (△), 55 wt % (□), and 58 wt % (◇).

(8)

where d and δ are the repeat distance and the thickness of membranes. Substituting eq 8 into eq 6, one obtains

demonstrates that the repeat distance at rest first rapidly increases with increasing temperature from 25 °C up to 45−50 °C and then levels off. The increment of the repeat distance becomes small at higher concentrations. These results are similar to those in the C16E7 system.38 Comparing Figure 8 with Figure 1, we can infer that the increase in the repeat distance with increasing temperature occurs near the lower transition temperature, whereas near the upper transition temperature, the repeat distance depends on the temperature only slightly. Therefore, the correlation between the increase in the repeat distance at rest and the existence of the lower transition may be general in the CnEm systems. In the case of multilamellar vesicles, the free energy per unit volume can be expressed as F = 4π (2K + K̅ )

2 9π 2 (kBT ) 64 K (d − δ)4

σeff =

3πkBT 8(d − δ)2

(9)

If we assume that onions can be formed only when the deformation energy becomes less than a critical value, the temperature dependence of F(R) and, hence, σeff is very important to discuss the conditions of onion formation. So, we have calculated the effective surface tension from the measured repeat distance at rest. Figure 9 shows temperature dependence of the effective surface tension at different concentrations. It can be seen from the figure that σeff rapidly decreases as the temperature increases from 25 to 40 °C, close to TL* and levels

(3)

where K and K̅ are the bending modulus and saddle splay modulus of multilayers and expressed in terms of κ and κ̅ as κ K= (4a) d κ K̅ = ̅ d

(4b)

where d is the repeat distance. Equations 4a and 4b indicate that the free energy decreases with increasing repeat distance. However, this is just due to the decrease in the density of bilayers and still cannot explain the lower transition. So other factors should be taken into account. Elastic Energy of Polyhedral Onions. Gulik-Krzywicki et al.4 have performed freeze-fracture electron microscopy

Figure 9. Temperature dependences of the effective surface tension obtained by using eq 9 and the observed repeat distance at rest at 47 wt % (○), 50 wt % (△), 55 wt % (□), and 58 wt % (◇). G

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eq 11 gives a clue to clarify necessary conditions for the onion formation. Effects of Shear Rate. The re-entrant lamellar/onion transition with increasing shear rate has been reported by Roux and co-workers.2,3 They have shown that the onions are transformed into an oriented lamellar state via a coexistence region when the shear rate exceeds a critical value. Recently, Gentile et al.43 have found onion formation in a C12E5/water system (40 wt % C12E5). In this system, onions are stable only in a shear rate range of 1−10 s−1 and the coexistence of lamellae and onions is observed after a long period of time (10 and 5 h at 20 s−1 and 40 s−1, respectively) at higher shear rates, suggesting the possibility of the re-entrant lamellar/onion transition with varying shear rate in the CnEm systems as well. Zipfel et al.60 have studied the influence of shear on lamellar phases in the system sodium dodecyl sulfate (SDS)/decanol/ water with a constant water content of 67.4% but different SDS-decanol ratios using SANS, SALS, birefringence, and rheology. They have shown that the onions are formed in a certain shear rate region (between 20 and 100 s−1) (i.e., the reentrant lamellar/onion transition with increasing shear rate). They have also shown that the onion formation occurs only when the mole fraction of decanol in the SDS/decanol mixtures (xc) is between 0.31 and 0.35. Therefore, the onions can be formed only in a closed region in the shear rate−xc diagram. It should be noted that the variation of xc may increase the saddle splay modulus of bilayers because they have shown that ribbonlike and porelike defects exist at rest for xc = 0.28−0.4. If the increase of xc corresponds to the increase in the temperature in the CnEm systems, it is possible that the onions are formed in a closed region in the temperature−shear rate diagram as well. In our previous study on the C16E7 system, we have made a temperature−shear rate diagram in the shear rate range of 0.05−30 s−1 at 48 wt %, which shows that the lower temperature limit of the onion state decreases with increasing shear rate from 0.05 to 1 s−1. As described above, the transition temperature in the C10E3 system increases with increasing shear rate from 1 to 10 s−1. These results suggest that onions are formed in a closed region not only in the temperature−concentration diagram but also in the temperature−shear rate diagram as is schematically shown in Figure 10 of the Supporting Information, although we did not observe the discontinuous change in the shear stress− shear rate curve, at least below 600 s−1 (see Figure 1 of the Supporting Information). To discuss the conditions of onion formation more generally, we need a three-dimensional (3D) dynamic phase diagram where the shear rate (or shear stress), temperature, and concentration are chosen as variables. In this 3D phase diagram, the boundary between the lamellar and onion states may form a closed surface. It should be noted that in the C16E7 system, the transition temperature from the defective lamellar phase (with water-filled defects) to the classical (defect-free) lamellar phase at 48 wt % is about 70 °C, which is close to the lamellar-to-onion transition temperature (the same may hold true for the C14E5 system at 50 wt %, as can be seen from the 1D SAXS pattern in Figure 3, panels a and c). As the surfactant concentration increases, however, the upper temperature limit of the defects decreases. On the other hand, the lamellar-to-onion transition temperature increases. Therefore, it can be deduced that the existence of defects is not directly correlated with the onion formation in our system, although the increase in the repeat distance with increasing temperature is due to the decrease in

off. If the onion size is constant, temperature and concentration dependences of the deformation energy F(R) are similar to those of σeff. This can explain why the onions can be formed above a specific temperature TL* and why TL* increases as the concentration increases. Near the TU*, however, σeff increases only slightly, suggesting that there is no correlation between σeff and the existence of the upper transition. Deformation energy may be stored in the vertices of polydedra as well. If we regard the vertex as a part of a sphere, the mean curvature of the vertex is 2/λ and the area of the curved part is λ2 per vertex. To consider the energy of both edges and vertices, however, we should estimate each contribution more quantitatively. In the appendix, we have shown that the deformation energy of a polyhedral onion can be expressed as F(R ) = NedgepR2σeff θ∞3 + Nvertex(2K + K̅ )qπRθ∞2

(10)

where Nedge and Nvertex are the number of edges and vertices of a polyhedron, 2θ∞ is the angle between adjacent planes, and p and q are numerical constants in the order of unity. Inserting eqs 2a and 2b,4a and 4b, and 9 into eq 10, we obtain F (R ) 3π R2 = Nedgepθ∞3 8 (d − δ)2 kBT ⎞R 2κ ⎛ κ + Nvertexqπθ∞2 m ⎜2 + m̅ − δH0m⎟ kBT ⎝ κm ⎠d

(11)

When the onion radius is constant, the first term in eq 11 rapidly decreases with increasing temperature and levels off similarly to σeff (see Figure 9), whereas the second term monotonously increases with increasing temperature due to the decrease in the spontaneous curvature of monolayers H0m, although the bending elasticity of monolayers κm may slightly decrease. Thus, the free energy F(R) should take a minimum with increasing temperature (see Figure 10). As the concentration increases, F(R) increases due to the decrease of d. Because F(R) is always positive, the transition to onions does not occur at rest. Under the shear flow, however, it is possible that the onions can be formed only in the temperature range where F(R) is below a threshold. Although we should take into account the effects of shear flow for more detailed discussion,

Figure 10. Schematic temperature dependence of the deformation energy of an onion based on eq 11. The dashed line and the dasheddotted line are the contributions from the edges and vertices, respectively. Because the deformation energy F(R) is always positive, the transition to the onions does not occur at rest. Under the shear flow, however, it is possible that the onions can be formed only in the temperature range where F(R) is below a threshold (the dotted line). H

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the fraction of defects. Zipfel et al.60 also conclude that the onion formation is not directly linked to the ribbonlike and holelike defects. Transition Processes. Roux and co-workers have also discussed the mechanism for the lamellar-to-onion transition.2,7 Even in a perfectly oriented lamellar phase, there are many dislocations in order to fill up the gap of the shear apparatus, which is not uniform in the scale of the lamellar spacing. When the shear rate exceeds a critical value (γ̇C), the dislocation cannot follow the flow, leading to the undulation instability. Wunenburger et al.61 have estimated γ̇C based on this idea and the theoretical results of Oswald et al.62,63 However, the calculated γ̇C (from 10−4 to 10−3 s−1) is much smaller than the observed ones (1−10 s−1). Zilman and Granek64 have considered that the coupling of the short wavelength membrane undulations with the shear flow generates an effective lateral pressure due to the fixed geometry of the macroscopic lamellar. Above a critical shear rate, the lamellar buckles into a harmonic shape modulation with the wavenumber, qC. Assuming that the lamellar breakups into onions at this limit of stability, the critical shear rate for the formation of onions is predicted to scale as γ̇C ∝ d−5/2D−1/2 (D: gap spacing), which is close to the experimentally determined power law γ̇C ∝ d−3 reported by Diat et al.2 They have also predicted the scaling relation of the wavenumber as qC ∝ (γ̇d/ D)1/3, which has been confirmed experimentally by Courbin et al.,23,24 assuming that qC is inversely proportional to the onion size determined from SALS. However, the absolute value of the critical shear rate γ̇C predicted by Zilman and Granek is about 3 × 103 s−1 for typical values of parameters (D = 1 mm, d = 10 nm, η = 3 mPa s, and κ = kBT (kB and T are the Boltzmann’s constant and absolute temperature, respectively) which is about 103 times larger than the values usually observed. Mallow and Olmsted65 have modeled the flow as an effective anisotropic tension, which decreases the compression modulus. They consider two cases; if new layers can be generated by permeation or defects, decrease in lamellar spacing occurs. On the other hand, if the system cannot change the number of layers, or the process is very slow, an instability is induced for large enough tension above σC ∝ (dD)−1, which produces either cylinders or onions. Using the relation σC ≅ η γ̇C d, they predicted the critical shear rate as γ̇C ∝ d−2D−1 ∝ ϕ2D−1, a slightly different power law from that obtained by Zilman and Granek. However, the absolute values are about 2 × 104 s−1 for the same parameter values as described earlier, which is again 104 times larger than experimental values. Thus, despite these intensive theoretical efforts, the mechanism of the lamellar-to-onion transition is still unclear. In the present study, we have investigated the change in the lamellar orientation in the lamellar-to-onion and onion-tolamellar transition processes for both the lower and the upper transitions. In the lamellar-to-onion transition, further enhancement of the lamellar orientation to the velocity gradient direction occurs, followed by the enhancement of the orientation to the neutral direction. As can be seen from Figure 6, panels a and b, temperature dependence of the shear stress and the peak intensity for each direction is independent of the heating rate when the heating rate is slow enough. Moreover, the sequence of the change in the lamellar orientation is reversible for the temperature. These results suggest that the change in the peak intensity for each direction corresponds not to the transient structures but to the steady state determined by the temperature and shear rate.

In accordance with the model of Zilman and Granek, the orientation of lamellae for the velocity gradient direction should be enhanced when the effective lateral pressure approaches the critical value just before the coherent buckling occurs. The abrupt increase in the peak intensity for the velocity gradient direction is consistent with this prediction. After the coherent buckling, the lamellar orientation for the gradient direction should be suppressed. Instead, in the case of the stripe buckling with the wave vector of the undulation in the neutral direction, the orientation to the neutral direction is expected to be enhanced. The abrupt decrease in the velocity gradient peak and the increase in the neutral peak are again in accordance with these predictions. However, it is still unclear how the buckling structure is transformed to onions. The enhancement of the orientation to the neutral direction before formation of onions has also been reported before us by Richtering and co-workers19,20 in C10E3/water system by using small-angle neutron scattering (SANS). First,19 they have attributed it to the formation of multilamellar cylinders as intermediate structures between lamellae and onions and later20 to either multilamellar cylinders or a coherent stripe buckling proposed by Zilman and Granek. However, the further enhancement of the orientation to the velocity gradient direction just before that to the neutral direction has not been reported except for our previous paper.40 This may be due to the fact that their experiments have been performed under constant temperature and shear rate. In our previous study, we made rheo-SAXS experiments at a constant temperature of 73 °C and a shear rate of 1 s−1. In this case, the intensity in the gradient direction decreases rapidly and the maximum of the intensity only for the neutral directions is observed after 500 s (500 strain units), which is much shorter than in the temperature-scan and the shear-rate scan experiment [i.e., after about 3 000 s (∼10 000 strain unit)]. This indicates difficulty in observing the intensity maximum of the velocity gradient direction before the maximum of the neutral direction in the experiments at constant temperature and shear rate. On the other hand, in the temperature scan and shear rate scan experiments, we can control the transition rate, which enabled us to observe the enhancement of the lamellar orientation to the velocity gradient direction before that to the neutral direction. It should be noted that the theory of Zilman and Granek concerns only the lamellar-to-onion transition process. However, the present results suggest that their model can be applied to the onion-to-lamellar transition as well. Le et al.21 have reported temperature scan experiments (0.3 K/min) on the C10E3 system, by using SANS at a constant shear rate. In the transition region at 100 s−1, they have obtained the fraction of onions, assuming that the 2D SANS pattern in this region is a superposition of those for onions and lamellae. It has been shown that the transition is reversible for the temperature and that the fraction is monotonously changed both in the heating and cooling processes, which indicates that there is no singular change in the intensity in each direction at 100 s−1. They have also reported 2D SANS patterns in the radial and tangential configurations at 10 s−1 and at several temperatures. These 2D patterns also appear to be changed monotonously, although temperature dependence of the peak intensity is not shown. Medronho et al.31 have studied the transition processes in the C10E3 system by using deuterium NMR spectroscopy under shear as a function of time at several temperatures and shear rates. They have shown that the transition is continuous when I

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onions are formed, starting from the oriented lamellar phase. The initial time evolution of the quadrupolar splitting is consistent with the formation of multilamellar cylinders. When onions are transformed into an oriented lamellar phase, on the other hand, a discontinuous nucleation-and-growth process with a coexistence region is observed. In the C14E5 system, however, it is difficult to find such a difference as can be seen from Figures 6 and 7. In the NMR experiments of Medronho et al., first the temperature and the shear rate are set to those corresponding to the lamellar (or onion) state. Then, after stopping the shear, the temperature is changed to that corresponding to the onion (lamellar) state. After that, the shear is applied again and NMR spectra are collected. On the other hand, as described above, we have changed the temperature little by little under shear. This may cause the discrepancy between their conclusion and ours.

Figure 11. Wall of smectic liquid crystals.

5. CONCLUSIONS We have found for the first time the reentrant lamellar/onion (lamellar-onion-lamellar) transition with varying temperature under constant shear rate using Rheo-SAXS for the C14E5/ water system. The results can be summarized as follows: (1) The C14E5/water system exhibits the lamellar phase at rest in a wide temperature range from 15 to 75 °C. (2) The onion state exists only in a closed region in the temperature−concentration diagram at a constant shear rate. The C14E5 system has more general features than other CnEm systems and thus is considered to be useful in the investigation of the conditions for the onion formation, as well as the transition mechanisms. (3) We have investigated the change in the lamellar orientation in the lamellar-to-onion and onion-to-lamellar transition processes, both for the lower and upper transitions. For all the four kinds of transition processes, the following change in the lamellar orientation is observed: lamellar state (oriented to the velocity gradient direction) ↔ further enhancement of the orientation to the velocity gradient direction ↔ enhancement of the orientation to the neutral direction ↔ onion state. (4) The increase in the repeat distance at rest with increasing temperature is necessary for the existence of the lower transition. (5) Expression of the deformation energies stored in the edges and vertices of polyhedral onions has been derived. Although we should take into account the effects of shear flow for a more detailed discussion, this expression gives a clue for the clarification of necessary conditions for the onion formation.

width of the walls and approximate the radius of the curvature of the walls by (see Figure 11) rw =

(A2)

Then, the deformation energy of walls per unit area is 2 2 Fwall F 1 ⎛ θ ⎞ 2λ = 2w wall = K ⎜ ∞ ⎟ = A V 2 ⎝ λ ⎠ θ∞

KB̅ θ∞3 = σeff θ∞3 (A3)

Equation A3 differs from the more rigorous expression,58 Fw/A = (2/3)σeffθ∞3, only by the numerical factor 2/3. Because the area of a plane of stacked edges is proportional to R2, we obtain Fwall 2 pR = pR2σeff θ∞3 (A4) A where p is a proportionality constant depending on the geometry of the polyhedron. We use the same approximation as eq A3 for the vertex part. Then, the deformation energy per volume is Fedge(R ) =

2 ⎛ θ 2 ⎞2 ⎛ 1 ⎞2 Fvertex 1 ⎛2⎞ = K ⎜ ⎟ + K̅ ⎜ ⎟ = (2K + K̅ )⎜ ∞ ⎟ V 2 ⎝ rw ⎠ ⎝ rw ⎠ ⎝ λ ⎠

(A5)

The surface area of the curved part at the vertex can be approximated by πw2, and the distance between the center and the vertex is set to qR, where q is another proportionality constant. Then, we have

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APPENDIX We consider the wall in the smectic liquid crystals. The halfwidth of the wall is given by58 w = l/θ∞

w w λ ≈ = 2 sin θ∞ θ∞ θ∞

Fvertex =

Fvertex 2 πw qR = (2K + K̅ )πqRθ∞2 V

(A6)

The total deformation energy per onion is

(A1)

F(R ) = NedgeFedge(R ) + NvertexFvertex(R )

where 2θ∞ is the angle between two layers on both sides of the walls (see Figure 11). In eq A1, θ∞ is assumed to be very small. The expression for larger θ∞ has been derived by Blanc et al.66 According to their equation, however, the error is only a percent, even for θ∞ = π/6. Panizza et al.57 have derived eq 5, assuming that θ∞ = 1 rad and the curvature of the wall is 1/λ because their interest is the relation between the elastic modulus and the onion radius. To compare the contributions from stacked edges and stacked vertices to the free energy, however, the θ∞ term should be incorporated more quantitatively. So we use eq A1 for the

= NedgepR2σeff θ∞3 + Nvertex(2K + K̅ )qπRθ∞2

(A7)

where Nedge and Nvertex are the number of edges and vertices of a polyhedron. In the case of a dodecahedron for example, Nedge = 30, Nvertex = 20, and θ∞ = 31.717° = 0.5536 rad. If we consider a sphere with the same volume as the dodecahedron, the relation between the length of the edges (a) and the radius of sphere (R) is given by a = 0.8176 R. Then the proportionality constants are given by p = 0.4375 and q = 1.1457. Thus, we obtain J

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(8) Richtering, W. Rheology and Induced Structures in Surfactant Solutions. Curr. Opin. Colloid Interface Sci. 2001, 6, 446−450. (9) Mortensen, K. Structural Studies of Lamellar Surfactant Systems under Shear. Curr. Opin. Colloid Interface Sci. 2001, 6, 140−145. (10) Berni, M. G.; Lawrence, C. J.; Machin, D. A Review of the Rheology of the Lamellar Phase in Surfactant System. Adv. Colloid Interface Sci. 2002, 98, 217−243. (11) Bergenholtz, J.; Wagner, N. J. Formation of AOT/Brine Multilamellar Vesicles. Langmuir 1996, 12, 3122−3126. (12) Weigel, Lauger, J.; Richtering, W.; Lindner, P. Anisotropic Small Angle Light and Neutron Scattering from a Lyotropic Lamellar Phase under Shear. J. Phys. II 1996, 6, 529−542. (13) Müller, S.; Borschig, C.; Gronski, W.; Schmidt, C.; Roux, D. Shear-Induced States of Orientation of the Lamellar Phase of C12E4/ Water. Langmuir 1999, 15, 7558−7564. (14) Hoffmann, H.; Bergmeier, M.; Gradzielski, M.; Thunig, C. Preparation of Three Morphologically Different States of a Lamellar Phase. Prog. Colloid Polym. Sci. 1998, 109, 13−20. (15) Escalante, J. I.; Gradzielski, M.; Hoffmann, H.; Mortensen, K. Shear-Induced Transition of Originally Undisturbed Lamellar Phase to Vesicle Phase. Langmuir 2000, 16, 8653−8663. (16) Soubiran, L.; Staples, E.; Tucker, I.; Penfold, J.; Creeth, A. Effects of Shear on the Lamellar Phase of a Dialkyl Cationic Surfactant. Langmuir 2001, 17, 7988−7994. (17) Léon, A.; Bonn, D.; Meunier, J.; Al-Kahwaji, J.; Greffer, O.; Kellay, H. Coupling between Flow and Structure for a Lamellar Surfactant Phase. Phys. Rev. Lett. 2000, 84, 1335−1338. (18) Le, T. D.; Olsson, U.; Mortensen, K. Topological Transformation of a Surfactant Bilayer. Phys. B (Amsterdam, Neth.) 2000, 276−278, 379−380. (19) Zipfel, J.; Nettesheim, F.; Lindner, P.; Le, T. D.; Olsson, U.; Richtering, W. Cylindrical Intermediates in a Shear-Induced Lamellarto-Vesicle Transition. Europhys. Lett. 2001, 53, 335−341. (20) Nettesheim, F.; Zipfel, J.; Olsson, U.; Renth, F.; Linder, P.; Richtering, W. Pathway of the Shear-Induced Transition between Planar Lamellae and Multilamellar Vesicles as Studied by TimeResolved Scattering Techniques. Langmuir 2003, 19, 3603−3618. (21) Le, T. D.; Olsson, U.; Mortensen, K.; Zipfel, J.; Richtering, W. Nonionic Amphiphilic Bilayer Structures under Shear. Langmuir 2001, 17, 999−1008. (22) Le, T. D.; Olsson, U.; Mortensen, K. Packing States of Multilamellar Vesicles in a Nonionic Surfactant System. Phys. Chem. Chem. Phys. 2001, 3, 1310−1316. (23) Courbin, L.; Delville, J. P.; Rouch, J.; Panizza, P. Instability of a Lamellar Phase under Shear Flow: Formation of Multilamellar Vesicles. Phys. Rev. Lett. 2002, 89, 148305−1−4. (24) Courbin, L.; Pons, P.; Rouch, J.; Panizza, P. How do ClosedCompact Multi-lamellar Droplets Form under Shear Flow? Europhys. Lett. 2003, 61, 275−281. (25) Courbin, L.; Panizza, P. Shear-Induced Formation of Vesicles in Membrane Phase: Kinetics and Size Selection Mechanisms, Elasticity versus Surface Tension. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2004, 69, 021504−1−021504−12. (26) Fritz, G.; Wagner, N. J.; Kaler, W. W. Formation of Multilamellar Vesicles by Oscillatory Shear. Langmuir 2003, 19, 8709−8714. (27) Oliviero, C.; Coppola, L.; Gianferri, R.; Nicotera, I.; Olsson, U. Dynamic Phase Diagram and Onion Formation in the System C10E3/ D2O. Colloids Surf., A 2003, 228, 85−90. (28) Medronho, B.; Fujii, S.; Richtering, W.; Miguel, M. G.; Olsson, U. Reversible Size of Shear-Induced Multi-lamellar Vesicles. Colloid Polym. Sci. 2005, 284, 317−321. (29) Fujii, S.; Richtering, W. Size and Viscoelasticity of Spatially Confined Multilamellar Vesicles. Eur. Phys. J. E: Soft Matter Biol. Phys. 2006, 19, 139−148. (30) Medronho, B.; Miguel, M. G.; Olsson, U. Viscoelasticity of a Nonionic Lamellar Phase. Langmuir 2007, 23, 5270−5274. (31) Medronho, B.; Shafaei, S.; Szopko, R.; Miguel, M. G.; Olsson, U.; Schmidt, C. Shear-Induced Transitions between a Planar Lamellar

F(R ) = 30 × 0.4375 × 0.1697R2σeff + 20 × 1.1031(2K + K̅ )R = 2.227R2σeff + 22.06(2K + K̅ )R

(A8)

It should be noted that values of p and q (in the order of unity) and the ratio Nedge/Nvertex (1.5 for a dodecahedron) depend on geometry only slightly.

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ASSOCIATED CONTENT

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AUTHOR INFORMATION

S Supporting Information *

Shear rate dependence of shear stress at 50 wt % and 50 °C, evolution of shear stress versus strain with a stepwise increase in temperature in the experiments shown by closed circles in Figures 2c, 4a, 6c, and 7c evolution of shear stress versus strain at constant temperature and shear rate shown by open circles in Figures 2c and 4a, evolution of shear stress and the intensities of the SAXS peak versus strain in the experiments shown in Figure 6 (panels a and b), schematic temperature-shear rate diagram at 50 wt %. This material is available free of charge via the Internet at http://pubs.acs.org. Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Mr. Yukihiro Torii for measuring the shear rate dependence of the shear stress. This work was supported by MEXT KAKENHI, Grant 18068016 (Grant-in-Aid for Scientific Research on Priority Area “Soft Matter Physics”) and JSPS KAKENHI, Grant 23340124 [Grant-in-Aid for Scientific Research (B)]. The rheo-SAXS experiments have been performed on the BL15A and 6A at Photon Factory under the approval of the Photon Factory Program Advisory Committee (Proposal 2011G589) and on the BL40B2 at Spring-8 under the Priority Program for Disaster-Affected Quantum Beam Facilities (Proposals 2011G589 and 2011A1921). We thank Dr. Nobutaka Shimizu in Photon Factory and Dr. Noboru Ohta in JASRI at Spring-8 for assistance of the experiments. We also thank Prof. Rony Granek and Prof. C.-Y. D. Lu for valuable discussions.

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REFERENCES

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