Smectic layer undulations

Fig. 17. 2-dimensional cut of the undulation local order schematic picture. Only a part of the smectic layer undulates, i.e. the hackhones and the spacers. The layers undulations induce mounds and wells regions where the spacers adopt a splay-like configuration... 

Keywords Block copolymers Director Hydrodynamics Layer normal Layered systems Liquid crystals Macroscopic behavior Multilamellar vesicles Onions Shear flow Smectic A Smectic cylinders Undulations... 

Smectic A liquid crystals are known to be rather sensitive to dilatations of the layers. As shown in [34, 35], a relative dilatation of less than 10-4 parallel to the layer normal suffices to cause an undulation instability of the smectic layers. Above this very small, but finite, critical dilatation the liquid crystal develops undulations of the layers to reduce the strain locally. Later on Oswald and Ben-Abraham considered dilated smectic A under shear [36], When a shear flow is applied (with a parallel orientation of the layers), the onset for undulations is unchanged only if the wave vector of the undulations points in the vorticity direction (a similar situation was later considered in [37]). Whenever this wave vector has a component in the flow direction, the onset of the undulation instability is increased by a portion proportional to the applied shear rate. 

Consequently, a parallel alignment of smectic layers is linearly stable against undulations even if the perpendicular alignment might be more preferable due to some thermodynamic considerations. As we have shown in Fig. 8, this rigorous result of standard smectic A hydrodynamics is weakened in our extended formulation of smectic A hydrodynamics. When the director can show independent dynamics, an appropriate anisotropy of the viscosity tensor can indeed reduce the threshold values of an undulation instability. 

We need to explain why we only observe the 102 diffuse spot. We could have expected that the 101 diffuse spot should be stronger than the 102 . This can be explained by considering that the undulations only affect a part of the layer so that the structure factor governing this modulation will be different from that governing the smectic reflection intensities. For simplicity, the electron density profile of the smectic layer may here be represented by step functions (Fig. 16). 

Theoretical developments in the early 1980s showed that the nonlinear interaction of thermally excited layer undulations, which as we have seen have large amplitudes because of the Peierls-Landau instability, leads to interesting new effects in the hydrodynamics of smectic A at small wavevector and frequency. We present below a very brief outline of the physical arguments involved. ... 

A noteworthy aspect of the above solution is that it does not involve either lattice dilatation du/dz or layer undulation V n. Therefore, within the approximations of the linear theory considered here, screw dislocations in smectic A have no self energy (apart from the core), nor do they interact amongst themselves. In this respect they are entirely different from screw dislocations in crystals. 

In the experiment, it is possible to create a dilatation of the smectic layers with piezoelectric drivers. Evidently, an increase of the interlayer distance would cost a lot of energy. Instead, at a certain critical dilatation Xc = Tiks/d, where Xs =, a wave-like or undulation distortion is observed as illustrated by... 

In smectic A liquid crystals the only allowed deformation is specific undulation of the smectic layers, such that interlayer distance is kept constant and the director remains normal to the layer. According to [31] this deformation imposes the following limitation to the director field ... 

Dynamic processes that can be investigated by NMR include both the motions of individual molecules, e.g., conformational dynamics and molecular rotations, and collective motions, e.g., director fluctuations in nematic systems, layer undulations in smectic systems, or density modulations in columnar phases of discotic systems. Self-diffusion can be measured by NMR relaxation or field gradient methods, as discussed in Sec. 13 of Chap. VII of this Volume. Table 1 gives an overview of the time scales accessible by the most common experimental techniques and examples of the type of motion that can be studied. 

In these copolymers, the segregation of two components into the bilayer really takes place however, the formed phase was a dissipative (frustrated) structure as can be seen from the X-ray diffraction pattern in Fig. 9.22a. In addition to the ordered position of the mesogen in the smectic layer cycle, there is a periodic density fluctuation from the undulations parallel to the layers. Figure 9.22b expresses the structural model. The undulations of the layer are due to a reverse domain structure of the polar structure of Fig. 9.21. We expected it to emerge in one... 

X-ray measurements also clarified that B7 materials have at least two different phase structures. Some of them (for example, the first B7 material [25]) have distinct sharp peaks in the low angle range [16] which is characteristic to columnar structure. Other B7-type materials (we denote them by Bl ) have strong commensurate reflections -which indicate a layer structure - and small incommensurate satellites, which are very close to each other and hard to resolve with normal X-ray technique. These peaks were attributed to a one-dimensional undulation of the smectic layers with... 

The sinusoidal undulation at the boundary is seen to diminish exponentially across the smectic layers in the 2 -direction over a distance characterised by the penetration length . 

Figure 6.11 A schematic diagram of the anticipated undulations of the smectic layers under the influence of a magnetic field for the geometry introduced in Fig. 6.10. As H increases through the value defined by equation (6.177), the smectic layers will begin to distort. For H < He no layer undulations occur, while for H > He the layers will undulate or distort.

The addition of a layer compression energy Wcomp such as those of the forms introduced at equations (6.147) and (6.148), can also be added to w. Some preliminary theoretical results involving the onset of layer undulations in a Helfrich-Hurault transition in SmC using such an energy obtained, for example, via (6.147), (6.299), (6.301) and (6.302), has been reported by Stewart [263]. Also, the smectic layer compression constant B has been measured for various materials that exhibit SmA and SmC phases see the comments and references on page 284. 

The role of permeation has not been mentioned in this Chapter. This effect occurs when there is a mass transport through the structure [110, p.413]. At this stage, it would appear that an additional equation or term is perhaps needed as a supplement to the theory presented here in order to describe this phenomenon. Such a term for smectics was first discussed by Helfrich [123] and later by de Gennes [108], and some details can be found in de Gennes and Frost [110, pp.435-445] for the case of SmA liquid crystals. The modelling of dynamics of layer undulations has also been carried out by some authors. Ben-Abraham and Oswald [14] and Chen and Jasnow [39] have examined dynamic aspects of SmA undulations using models based on the static theory described in Section 6.2.6 which incorporate flow and the influence of permeation. Experimental observations of a boundary layer in permeative flow of SmA around an obstacle have been reported by Clark [48]. Some more recent experimental and theoretical results involving permeation with compression and dilation of the smectic layers in a flow problem around a solid obstacle where there is a transition from SmA to SmC have been presented by Walton, Stewart and Towler [277] and Towler et al [269]. 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

In the previous sections we have shown that the inclusion of the director of the underlying nematic order in the description of a smectic A like system leads to some important new features. In general, the behavior of the director under external fields differs from the behavior of the layer normal. In this chapter we have only discussed the effect of a velocity gradient, but the effects presented here seem to be of a more general nature and can also be applied to other fields. The key results of our theoretical treatment are a tilt of the director, which is proportional to the shear rate, and an undulation instability which sets in above a threshold value of the tilt angle (or equivalently the shear rate). 

Experiments by Muller et al. [17] on the lamellar phase of a lyotropic system (an LMW surfactant) under shear suggest that multilamellar vesicles develop via an intermediate state for which one finds a distribution of director orientations in the plane perpendicular to the flow direction. These results are compatible with an undulation instability of the type proposed here, since undulations lead to such a distribution of director orientations. Furthermore, Noirez [25] found in shear experiment on a smectic A liquid crystalline polymer in a cone-plate geometry that the layer thickness reduces slightly with increasing shear. This result is compatible with the model presented here as well. 

Alternatively, if one dilates a smectic stack by increasing its thickness by an amount Sh > 27t X, then the sample will prefer to bend the layers in an undulational instability (Rosenblatt et al. 1977 Ostwald and Allain 1985) in order to restore the lamellar spacing to its preferred value (see Fig. 10-29d). Note that the increase in thickness 8h required to produce this instability is independent of the initial thickness h of the stack. Hence for a macroscopic sample of thickness, say, h = 60 p,m, the strain Sh/h required to induce the undulational instability is extremely small, 8h/h IzrXfh 10-4. Thus smectic monodomains are extremely delicate and can easily be disrupted by mechanical deformation. 

Figure 10.29 Response of an aligned smectic to layer dilation, (a) Initial equilibrium sample, (b) For a very small dilation Sh < Ink, the layer spacing simply increases, (c) A uniform rotation of the layers decreases the spacing toward that of equilibrium, but doesn t satisfy the boundary conditions, (d) Hence, the sample undergoes an mdulational instability, which also narrows the layer spacing while satisfying homeotropic boundary conditions, (e) For a large enough dilation, the undulation instability leads to formation of parabolic focal conic defects. (From Rosenblatt et al. 1977, with permission from EDP Sciences.)...

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