. At high shear rate, the orientation is similar to that at low shear rate but defects disappear. They also gave simple explanation on the power law behavior of the onion size by considering balance between shear stress and elastic stress of the bilayer membranes. Although they could identify the structure of the onion phase, detailed transition mechanism was not mentioned.
In order to understand the onion formation mechanism from planar lamellar phase, Nettesheim et al. traced the transition process by using small-angle neutron scattering under shear (Rheo-SANS) and found existence of intermediate structure as schematically shown in Fig. 4.1 [3]. They identified the intermediate structure with multilamellar cylinder structure or coherently buckled lamellae. Medronho et al. recently presented the experimental results supporting the multilamellar cylinder structure on the basis of the anisotropic diffusion of water measured by Rheo-NMR [4]. On the other hand, Kato et al. very recently presented that the coherently buckled lamellae are plausible compared to the cylinder structure by observing a small-angle X-ray scattering profiles under shear (Rheo-SAXS) [5]. Although new experimental results have been steadily accumulated, the comprehensive studies on the other physical quantities such as critical condition and membrane flexibility necessary for the onion phase formation are still lacking.
Fig. 4.1
Schematic diagram of (a) planar lamellar phase, (b1) multilamellar cylinder structure and (b2) coherently buckled lamellae, and (c) onion phase
4.1.2 Other Transitions of Layered System Under Shear
Shear-induced structural transition is an unsolved topic in a soft matter science. It has been widely reported that many different surfactant lamellar systems show the shear-induced onion phase formation [1–10]. In addition to the lyotropic lamellar systems, thermotropic smectic A phase , more studied layered system rather than lyotropic system, also shows a unique shear-induced transition as presented by Safinya et al. and Panizza et al. [11, 12].
Figure 4.2 presents a stress as a function of shear rate performed for one component material 8CB, one of the most known thermotropic smectic phases. The flow curve shows a characteristic feature of the shear-induced transition between shear-thinning region (described as I) and Newtonian region (II). Panizza et al. also performed small-angle X-ray scattering (SAXS) measurements in two regions I and II. They identified this rheological transition with layer orientation transition from multilayered cylinder aligned in the flow direction (I) to the layer with layer normal along the vortex direction, thus a perpendicular orientation (II). Very recently, we also systematically studied the layer orientation transition as a function of temperature and shear stress for the same 8CB smectic A phase and presented dynamic orientation diagram in detail [13]. Interestingly, the thermotropic smectic A phase never shows the onion phase formation. One of the distinctions between the onion phase formation and layer orientation transition is the magnitude of the critical shear rate. Layer orientation transition appears at high shear rate region roughly at 1,000 s−1, while the onion phase is obtained at rather low shear rate such as 1 s−1. Salient difference in the critical shear rate suggests that the instability triggered by shear flow has different physical origin. However, it has not been discussed from what the difference in these transitions comes. It has not been clear whether these transitions have the same physical origin or different physical background. In this chapter, we will mainly focus on the shear-induced onion phase formation. We will not discuss on the layer orientation transition here. However, one can find some articles on the layer orientation transition of the thermotropic smectic A phase and lyotropic lamellar phase [11–13].
Fig. 4.2
Stress vs. shear rate for thermotropic 8CB smectic phase and schematic diagrams of multilayered cylinder structure and perpendicular orientation expected for shear rate regions I and II, respectively
4.1.3 Simple Explanation of Models on Onion Formation
Two models are generally suggested to explain the origin of the onion phase formation. One explanation is described by Zilman and Granek [14]. Their model considers the coupling of the short wavelength undulation of the lamellar phase with the shear flow. Short wavelength undulations produce a long-range repulsive force, the undulation force, which is responsible for the stability of the lamellar phase . Reduction of the excess area and suppression of membrane undulations due to shear flow thus generate an effective dilatation . As the dilatation becomes higher than the critical shear rate, the coherent buckling of the lamellar membranes appears, and the onion phase formation is achieved. This proposed mechanism that attracted many scientists engaging in the shear-induced onion phase formation. However, one finds mismatch in the critical shear rate. The critical shear rate 
~1,000s−1 calculated on the basis of typical parameters such as a repeating distance of the lamellar membranes of 10 nm, a bending modulus of 
~ k B T and solvent viscosity, is much higher than that of experimentally obtained one 
~ 1-10 s−1. Although good agreement can be obtained by using the effective viscosity of the lamellar phase, the origin of the high viscosity is unclear. One of the reasons for such differences might be a contribution of defects.
~1,000s−1 calculated on the basis of typical parameters such as a repeating distance of the lamellar membranes of 10 nm, a bending modulus of ~ k B T and solvent viscosity, is much higher than that of experimentally obtained one ~ 1-10 s−1. Although good agreement can be obtained by using the effective viscosity of the lamellar phase, the origin of the high viscosity is unclear. One of the reasons for such differences might be a contribution of defects.The other mechanism mainly based on the work of Oswald considers the motion of defects [15, 16]. In layered systems, nonuniform gap in the sample cell leads to the formation of defects. In the presence of defects, the number of layers shows local inhomogeneity. This inhomogeneity can lead a dilatational and compressive strain normal to the layers. Dislocations in the lamellar phase are able to move with flow by permeation at low shear rate. Here, permeation is the diffusion of surfactant monomer and solvent across the membranes. However, the permeation process is too low to allow dislocations to flow at high shear rates. Therefore, a delay of dislocations against the shear flow generates a dilative and compressive strain. This nonuniform shear flow becomes the origin of the undulation instability and results in the onion phase formation. Qualitative explanation argued by Roux et al. is also similar to this idea [1, 2]. Interestingly, common origin of the instability in the layered systems is a dilative strain in spite of the inherent mechanism of the shear-induced onion phase formation. However, we should note that the dilative strain should also produce defects before the onion phase is formed. Eventually, we will also mention about the role of defects on the onion phase formation.
4.1.4 Defects and Dislocations
Lamellar phase always includes locally unregulated alignment of layers, i.e., defects . As typical defects in layered systems, two dislocations, edge and screw dislocations, and a texture with meso-scale size, focal conic domain, are known [17, 18]. Edge and screw dislocation s are linear defects. Since these dislocations never possess terminal, a pair of edge dislocations and screw dislocations with opposite signs usually form a dislocation loop as shown in Fig. 4.3.
Fig. 4.3
Schematic diagram of (a) edge and screw dislocations and dislocation loop in which two dislocations are paired and (b) focal conic domain type I (FCD-I), type II (FCD-II), and oily streak defects composed of FCD-I connected by edge dislocations
Formation of the dislocation loop accompanies unfavorable local compression deformation of the lamellar membranes. Thus, the increase in the dislocation loop density allows the strain energy to accumulate. Such accumulation of the strain energy is relaxed down by creating new texture, focal conic domains (FCDs). FCD can be easily obtained because of the constant layer repeating distance inside it, and this texture is actually often observed in lyotropic lamellar systems. Depending on the sign of the curvature of membranes, FCD is further classified into two species: FCD-I with torus-like deformation of membranes and FCD-II with onion-like structure. Because of the similarity of the structure, the onion phase itself is sometimes referred to as one kind of defect. One should also note that several FCD-Is are often connected in series by edge dislocations. The connected FCD-I is referred to as oily streak defects [17].
4.1.5 Smectic Rheology
Here, we propose a mesoscopic structure-mediated rheology in soft matter systems, called “structural rheology.” This is a new concept in the soft matter physics. One of such examples of the structural rheology is a smectic rheology in which the defects in the meso-scale structure significantly affect the rheology.
It is well recognized that the defects govern the rheological properties of the lamellar phase [19–22]. Oily streaks and dislocation loops mentioned above dominate the elasticity and the shear-thinning behavior of the lamellar phase, respectively. It has also been considered that defects are deeply correlated with the onion formation. Medronho et al. actually presented that the increase in the defect density is necessary as a pretransition of the onion phase formation by measuring the shear modulus [23]. The increase in the oily streak density under shear might be considered to be a pretransition. On the other hand, Dhez et al. qualitatively mentioned that the length of the screw dislocation determines the threshold of the shear-induced onion formation [24]. Although the importance of defects seems to be a common understanding [1, 2, 23, 24], nobody has mentioned how defects contribute to the shear-induced onion phase formation.
4.2 Shear-Induced Onion Formation in Complex Bilayer Lamellar Phase
4.2.1 Complex Bilayer Lamellar Phases
Complex bilayer lamellar phase is composed of nonionic surfactant, triethylene glycol monodecyl ether (C10E3), amphiphilic triblock copolymers, Pluronic supplied by BASF, and distilled water. Triblock copolymer, Pluronic, consists of two hydrophilic ethylene oxide (EO) blocks bounded to a central hydrophobic propylene oxide (PO) block. Hydrophilic EO chains at both sides have an identical chain length. In complex bilayer membranes, the hydrophobic PO block interpenetrates or stacks inside the hydrophobic region of the surfactant bilayers, and EO blocks decorate the membranes as presented by Masui et al. for nonionic surfactant C12E5/Pluronic/D2O systems [25]. Schematic diagram of the complex bilayer membrane is shown in Fig. 4.4.
Fig. 4.4
Schematic diagram of complex bilayer lamellar phase composed of surfactant bilayer and grafted triblock copolymer chains
In the triblock copolymer-decorated lamellar phase, on the one hand, several triblock copolymers having different degrees of polymerization of EO chains (N EO = 17, 27, and 37) are used, while the degree of polymerization of PO block is fixed at N PO = 60.
In the triblock copolymer-embedded lamellar phase, on the other hand, the degree of polymerization of PO chains is varied as N PO = 16, 30, 42, 59, and 69, while the degree of polymerization of EO block is fixed at N EO = 4, which is almost the same as the hydrophilic chain length of the surfactant.
The mole fraction of triblock copolymer in the surfactant bilayer X P is calculated by the following relation: X P = n P/n S + n P. Here, n P and n S are, respectively, the mole number of the triblock copolymer and surfactant. Total mole number n S + n P is fixed at 8.01 mmol, which corresponds to the weight fraction of the surfactant of 40 wt% in C10E3/H2O binary system.
The bending modulus of such polymer-grafted membranes depends on the size and a number density of the hydrophilic polymer chain as theoretically predicted by Hiergeist and Lipowsky [25, 26]. Varying these quantities, effects of the defect density and the bending rigidity on the shear-induced onion phase formation can be systematically studied [27].
4.2.2 Shear Viscosity
4.2.2.1 Triblock Copolymer-Decorated System
Figure 4.5 shows the shear rate dependence of the viscosity for C10E3/P-EO(37)/H2O systems with different mole fractions of X P = 0.2, 0.8, 1.0, and 1.2 %. Here, P-EO(37) expresses Pluronic with EO blocks of N EO = 37. For the comparison, the viscosity of the polymer-free (C10E3/H2O) system is also shown as open circle. The shear-induced onion phase formation of the polymer-free system accompanies a shear-thickening behavior [10]. Appearance of the shear thickening can be identified as the initiation of the onion phase formation for nonionic surfactant lamellar system [3, 10]. The shear-thickening region, however, corresponds to a transition region where a coexistence of the planar lamellar and the onion phases is expected, and the onion phase formation is completed in the shear-thinning region at the higher shear rate. A critical shear rate of the shear thickening in the polymer-free system is 
=3 s−1. In the polymer-grafted system, anomalous dependence of the critical shear rate 
on X P is observed.
=3 s−1. In the polymer-grafted system, anomalous dependence of the critical shear rate on X P is observed.Fig. 4.5
Viscosity vs. shear rate for C10E3/P-EO(37)/H2O system with different mole fractions of X P. Viscosity of polymer-free system (C10E3/H2O) is also plotted for the reference. Vertical dashed line corresponds to the critical shear rate of the polymer-free system. Solid lines corresponds to a slope of −1/3 [27]. Reproduced by permission of The Royal Society of Chemistry
At low X P(=0.2 %), 
shifts to lower shear rate indicating that the onion formation was easily achieved. On the contrary, remarkable shift of 
into high shear rate region is observed at higher X P(=1.0 %). For such systems, the first shear thinning at low shear rate followed by the shear thickening and the second shear thinning at high shear rate can be observed. The first shear thinning follows a power law relation of 
. Once X P exceeded the overlap concentration of EO chains in the water layer C* = 1.1 %, the shear thickening is never observed as can be seen for X P = 1.2 %. In this system, the same power law exponent −1/3 as the first shear thinning can be observed in a wide shear rate region. Qualitatively, similar behavior on the X P dependence of the flow curve is observed for other systems with different EO chain lengths [27]. The shear-induced onion formation is favored at lower mole fraction, while 
remarkably shifts into higher shear rate region at higher mole fraction. The EO block-decorating membranes have strong impact on the critical shear rate.
shifts to lower shear rate indicating that the onion formation was easily achieved. On the contrary, remarkable shift of into high shear rate region is observed at higher X P(=1.0 %). For such systems, the first shear thinning at low shear rate followed by the shear thickening and the second shear thinning at high shear rate can be observed. The first shear thinning follows a power law relation of . Once X P exceeded the overlap concentration of EO chains in the water layer C* = 1.1 %, the shear thickening is never observed as can be seen for X P = 1.2 %. In this system, the same power law exponent −1/3 as the first shear thinning can be observed in a wide shear rate region. Qualitatively, similar behavior on the X P dependence of the flow curve is observed for other systems with different EO chain lengths [27]. The shear-induced onion formation is favored at lower mole fraction, while remarkably shifts into higher shear rate region at higher mole fraction. The EO block-decorating membranes have strong impact on the critical shear rate.4.2.2.2 Triblock Copolymer-Embedded System
Figure 4.6 shows a typical shear rate dependence of the viscosity for triblock copolymer-embedded systems with different degrees of polymerization of N PO = 16, 30, 42, 59, and 69 at X P = 1 mol% instead of fixed EO block of N EO = 3. At low N PO (=16 and 30), 
is not remarkably influenced by embedded hydrophobic chain. As N PO is increased, on the contrary, significant shift of the critical shear rate into high shear rate region has appeared. Finally, at N PO =69, the shear thickening as the signature of the onion formation disappears. Similar tendency has been observed when the hydrophilic chain length N EO is increased [27]. In X P dependence, increasing X P makes the critical shear rate higher (for more information, please see [38]).
is not remarkably influenced by embedded hydrophobic chain. As N PO is increased, on the contrary, significant shift of the critical shear rate into high shear rate region has appeared. Finally, at N PO =69, the shear thickening as the signature of the onion formation disappears. Similar tendency has been observed when the hydrophilic chain length N EO is increased [27]. In X P dependence, increasing X P makes the critical shear rate higher (for more information, please see [38]).Fig. 4.6
Viscosity vs. shear rate for C10E3/P-PO(N PO)/H2O system with different mole fractions of X P
4.2.3 Structural Observations
Figures 4.7 summarizes typical small-angle light-scattering (2D-SALS) patterns of different systems at several shear rates. The data shown in Fig. 4.7 corresponds to results in Fig. 4.5. SALS pattern of the polymer-free system is markedly allowed to develop with the shear rate. In the beginning of the shear thinning (at 
= 10s−1), a four-leaves pattern, a typical feature of the shear-induced onion phase, is observed. As the shear rate is increased, a refinement of the onion structure yields the Bragg peak in addition to the four leaves. The Bragg peak is a representative of the shear-induced onion phase with monodispersed size distribution. In general, such pattern is always observed in the shear-thinning region [3, 28]. Kato et al. attributed the four leaves with the Bragg peak to closed-pack onions in FCC packing.[6].
= 10s−1), a four-leaves pattern, a typical feature of the shear-induced onion phase, is observed. As the shear rate is increased, a refinement of the onion structure yields the Bragg peak in addition to the four leaves. The Bragg peak is a representative of the shear-induced onion phase with monodispersed size distribution. In general, such pattern is always observed in the shear-thinning region [3, 28]. Kato et al. attributed the four leaves with the Bragg peak to closed-pack onions in FCC packing.[6].Fig. 4.7
SALS pattern obtained for (a) polymer-free system and for several systems with different EO chain length N EO at fixed mole fraction of X P = 0.2 % under several shear rates [27]. Reproduced by permission of The Royal Society of Chemistry
Grafting polymer chains onto the membrane remarkably affects the scattering profile as well as the flow curve depends on X P, N EO, and N PO. Increasing X P at fixed N EO (or N PO), or vice versa, disturbs the development of the four-leaves pattern and allows disappearance of the Bragg peak even in the shear-thinning region. Shear rate dependence of the scattering pattern for a system with N EO = 37 at X P = 0.4 % indicates no significant development of the onion structure. Weak shear effect on the onion radius was also observed for the hydrophobically modified polymer-grafted lamellar systems by Yang et al. [29]. As X P is increased, the four-leaves pattern eventually vanishes at X P = 1.2 %. The shear-thinning behavior at X P = 1.2 % in Fig. 4.5 is thus attributed to the lamellar phase but not to the onion phase.
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