Smectic cylinders

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

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

Instabilities under mechanical tension in a smectic cylinder", Phvs.. (Paris) 31, 1359 (1976). 

Flard spherocylinders (cylinders witli hemispherical end caps) were studied using computer simulations [118]. In addition to a nematic phase, such particles also display a smectic-A phase, in which tire particles are arranged in liquid-like layers. To observe tliis transition, ratlier monodisperse particles are needed. The smectic-A phase was indeed observed in suspensions of TMV particles [17]. 

Figure 8.2 Schematic representation of the molecular arrangement in nematic (N), smectic A (SmA) and smectic C (SmC) mesophases. A cylinder represents a molecule with rod-like shape.

Fig. 52 Sequence of LC phases formed by self-assembly of T-shaped bolaamphiphiles (e.g. compounds 182), depending on the size of the semiperfluorinated lateral chain (a,b) smectic phases, (c-h) polygonal LC cylinder phases, and (i-1) Lam phases [8]...

We note that earlier research focused on the similarities of defect interaction and their motion in block copolymers and thermotropic nematics or smectics [181, 182], Thermotropic liquid crystals, however, are one-component homogeneous systems and are characterized by a non-conserved orientational order parameter. In contrast, in block copolymers the local concentration difference between two components is essentially conserved. In this respect, the microphase-separated structures in block copolymers are anticipated to have close similarities to lyotropic systems, which are composed of a polar medium (water) and a non-polar medium (surfactant structure). The phases of the lyotropic systems (such as lamella, cylinder, or micellar phases) are determined by the surfactant concentration. Similarly to lyotropic phases, the morphology in block copolymers is ascertained by the volume fraction of the components and their interaction. Therefore, in lyotropic systems and in block copolymers, the dynamics and annihilation of structural defects require a change in the local concentration difference between components as well as a change in the orientational order. Consequently, if single defect transformations could be monitored in real time and space, block copolymers could be considered as suitable model systems for studying transport mechanisms and phase transitions in 2D fluid materials such as membranes [183], lyotropic liquid crystals [184], and microemulsions [185],... 

Behavior that is intermediate between that of a solid and that of a liquid is perhaps not surprising for a block copolymer with hexagonally ordered cylinders, since such a material has solid-like order in the two directions perpendicular to the cylinders and liquid-like order parallel to the cylinders. Similar behavior is observed in lamellar block copolymers, which has solid-like order in the direction normal to the lamellae and has liquid-like order in the other two directions. For lamellar block copolymers, solid-like behavior at low frequencies typically arises from the disrupting effect of defects, such as those present in smectic liquid crystals (see Section 10.4.8). 

Fig. 41 Schematic drawing of the bilayer structure of the smectic A phase of multipede 39. The 16 mesogenic units (cylinders and spheres which represent the cyano groups) per molecule are accommodated in the layers without the introduction of curvature in the packing of the mesogenic units together...

If the side chain liquid crystalline polymers goes into the smectic A phase from the nematic phase, the backbone chain is confined between two successive smectic layers, occasionally jumping into the neighboring layer gap. See Figure 2.30 where the cylinders denote side groups and thick lines represent backbones. The mean square end-to-end distances parallel and perpendicular to the director differs more than that in the nematic phase. 

Figure 4 Experimental smectic mesophase with cylinders at phospholipid-water...

Thermotropics may display various organized structures. The nematics (from the greek nematos=wire) are elongated cylinders which have one directional order and lie more or less parallel to one another (Fig.2b) Smectics are organized in layers where the director may have a variable orientation relative to that of the layer. Smectic A are perpendicular (Fig. 2c) and Smectic C tilted (Fig. 2d). The various structures of the layer lead to the distinction of 8 smectic phases (for details see Ref. 16-17). 

Fig. 5.4.1. (a) Smectic layers in concentric cylinders to form a myelin sheath with a singular line L along the axis (b) the cylinders are closed to form tori there are two singular lines, a circle and a straight line (c) the general case when the smectic layers form Dupin cyclides the circle becomes an ellipse and the straight... 

Fig. 6.4.6. Developable domains in the columnar phase, (a) The developable surface is degenerated into a straight line d common to the planes P. The columnar axes C form coaxial circles about 5. b) The developable surface Z) is a cylinder. The columns (or the layers in the case of smectic A) are a set of parallel and equispaced involutes of a circle, (c) A Reimann surface generated by half-tangents to a helix. The columns are normal to the half-tangents. (Bouligand. )...

Figure 13. Structure of the famous leaning Tower of Pisa. Note that the packing is also identical to that found in LC cylinder morphology with a smectic mes-ophase in the LC domain. The tower represents the cylinder microdomain and the columns represent meso-gens which are packed in a layer structure like the SmA phase.

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