Macroscopic director

For a removal attempt a molecule is selected irrespective of its orientation. To enhance the efficiency of addition attempts in cases where the system possesses a high degree of orientational order, the orientation of the molecule to be added is selected in a biased way from a distribution function. For a system of linear molecules this distribution, say, g u n ), depends on the unit vector u parallel to the molecule s symmetry axis (the so-called microscopic director [70,71]) and on the macroscopic director h which is a measure of the average orientation in the entire sample [72]. The distribution g can be chosen in various ways, depending on the physical nature of the fluid (see below). However, g u n ) must be normalized to one [73,74]. In other words, an addition is attempted with a preferred orientation of the molecule determined by the macroscopic director n of the entire simulation cell. The position of the center of mass of the molecule is again chosen randomly. According to the principle of detailed balance the probability for a realization of an addition attempt is given by [73]... 

On a molecular level the director is not rigorously defined, but the molecular director is typically considered to be the average long axis of the molecules, oriented along the macroscopic director with some order parameter less than one. This type of anisotropic order is often called long-range orientational order and, combined with the nonresonant optical properties of the molecules, provides the combination of crystal-like optical properties with liquidlike flow behavior characteristic of liquid crystals. 

Note 1 In DSM the Williams Kapustin) domains become distorted and mobile, and macroscopic director alignment is completely disturbed. 

The influence of the crosslinking density on the birefringence has been studied [11]. For weakly crosslinked materials the results obtained coincide with the previous ones while for strongly crosslinked liquid crystalline elastomers the birefringence is only weakly affected in the vicinity of the phase transition as can be seen in Fig. 17. We stress, however, that the strongly crosslinked LSCE are duromers for which there are no independent macroscopic director degrees of freedom anymore, which are characteristic for the nematic liquid crystalline phase in low molecular weight materials, in sidechain polymers and in weakly crosslinked liquid crystalline elastomers. 

Other more exotic types of calamitic liquid crystal molecules include those having chiral components. This molecular modification leads to the formation of chiral nematic phases in which the director adopts a natural helical twist which may range from sub-micron to macroscopic length scales. Chirality coupled with smectic ordering may also lead to the formation of ferroelectric phases [20]. 

Another example of the coupling between microscopic and macroscopic properties is the flexo-electric effect in liquid crystals [33] which was first predicted theoretically by Meyer [34] and later observed in MBBA [35], Here orientational deformations of the director give rise to spontaneous polarisation. In nematic materials, the induced polarisation is given by... 

Since P must remain normal to z and n, the polarization vector forms a helix, where P is everywhere normal to the helix axis. While locally a macroscopic dipole is present, globally this polarization averages to zero due to the presence of the SmC helix. Such a structure is sometimes termed a helical antiferroelectric. But, even with a helix of infinite pitch (i.e., no helix), which can happen in the SmC phase, bulk samples of SmC material still are not ferroelectric. A ferroelectric material must possess at least two degenerate states, or orientations of the polarization, which exist in distinct free-energy wells, and which can be interconverted by application of an electric field. In the case of a bulk SmC material with infinite pitch, all orientations of the director on the tilt cone are degenerate. In this case the polarization would simply line up parallel to an applied field oriented along any axis in the smectic layer plane, with no wells or barriers (and no hysteresis) associated with the reorientation of the polarization. While interesting, such behavior is not that of a true ferroelectric. 

Many other interesting examples of spontaneous reflection symmetry breaking in macroscopic domains, driven by boundary conditions, have been described in LC systems. For example, it is well known that in polymer disperse LCs, where the LC sample is confined in small spherical droplets, chiral director structures are often observed, driven by minimization of surface and bulk elastic free energies.24 We have reported chiral domain structures, and indeed chiral electro-optic behavior, in cylindrical nematic domains surrounded by isotropic liquid (the molecules were achiral).25... 

Note 2 In the case of a short pitch, when P is less than the wavelength X, the macroscopic extraordinary axis for the refractive index is orthogonal to the director. 

Turning now to those molecules whose shape can be approximated by oblate spheroids, one arrives at the discotic phase. Here the average of the normals to planes of the molecules corresponds to the director. A fluid phase in which these normals point in roughly the same direction over a macroscopic distance is said to be discotic. If this factor is the only degree of order, the material is said to be in the nematic discotic phase. If, in addition, the discs stack in regular columns, the material is said to be in the columnar discotic phase. Such structures have been discussed in Section 4.5.1. 

Consider a lamellar mesophase, being macroscopically aligned so that the symmetry axis, referred to as the director, has the same direction throughout the sample. If the transformation from the molecular coordinate system to the laboratory system is performed via the director coordinate system (D), Equation 2 reads... 

Comparison of experimental data with Equation 7 makes it possible to determine how the director is oriented with respect to the constraint (14) responsible for macroscopic alignment. 

For this discussion, several points should be stressed here. Most importantly, there is no polar order along the director in any known liquid crystal phase, including the C phase. Thus, functional arrays with large P along the director are not oriented along a polar axis in the FLC phase. This is our interpretation of the small of DOB AMBC and other FLC materials. There are other possible problems as well, however. For example, though DOBAMBC possesses substantial dipoles oriented normal to the director, it s observed macroscopic polarization (-0.009 D/molecule) is very small. This could be due to poor molecular orientation in the FLC phase, which in turn could represent a fundamental problem in design of FLCs for x<2). 

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

In the framework of irreversible thermodynamics (compare, for example, [31, 32]) the macroscopic variables of a system can be divided into those due to conservation laws (here mass density p, momentum density g = pv with the velocity field v and energy density e) and those reflecting a spontaneously broken continuous symmetry (here the layer displacement u characterizes the broken translational symmetry parallel to the layer normal). For a smectic A liquid crystal the director h of the underlying nematic order is assumed to be parallel to the layer normal p. So far, only in the vicinity of a nematic-smectic A phase transition has a finite angle between h and p been shown to be of physical interest [33],... 

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. 

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

FIG. 15.49 Orientation angle 6 between rodlike molecules, described by unit vector u and director n within one nematic domain of a macroscopically isotropic polydomain sample a denotes the size of the domain. From Beekmans, 1997. 

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