Liquid crystal alignment, theory

Starting from equation (12.16a), show that according to the version of the theory of liquid-crystal alignment presented in section 12.4.4 the minimum aspect ratio for the rods of a material that exhibits thermotropic liquid crystallinity is 5.44. 

Another peculiar property of LCPs is shown in Fig. 15.47, where the transient behaviour of the shear stress after start up of steady shear flow is shown for Vectra A900 at 290 °C at two shear rates. We will come back to this behaviour in Chap. 16 for lyotropic systems where this behaviour is quite common and in contradistinction to the transient behaviour of conventional polymers, as presented in Fig. 15.9. This damped oscillatory behaviour is also found for simple rheological models as the Jeffreys model (Te Nijenhuis 2005) and according to Burghardt and Fuller, it is explicable by the classic Leslie-Ericksen theory for the flow of liquid crystals, which tumble, rather than align, in shear flow. Moreover, it is extra complicated due to the interaction between the tumbling of the molecules and the evolving defect density (polynomial structure) of the LCP, which become finer, at start up, or coarser, after cessation of flow. 

The steady state is reached after several oscillations and the time of the minima and maxima may be scaled by qt, where q is the constant shear rate. As already said in Chap. 15, this behaviour is according to Burghardt and Fuller explicable by the classic Leslie-Ericksen theory for the flow of liquid crystals, which tumble, rather than align, in shear flow. Again it is far beyond the scope of this book to go into detail of this theory. 

D. Forster, Microscopic Theory of Flow Alignment in Nematic Liquid Crystals, Phys. Rev. Lett. 32 (1974) 1161. 

In the theoretical analysis of electrically driven pattern formation in nematics one deals only with the theoretical AC voltage Utheo, which drops over the nematic layer. Utheo differs, however, from the experimental voltage Uexp applied to the whole LC cell and recorded in experiments. Thus a quantitative comparison between experiments and theory is far from trivial as has been emphasized for instance by Krekhov et Typical liquid crystal cells consist of a nematic layer confined between ITO- or Sn02-coated glass plates covered with a thin film of an aligning polymer. As the polymer is a quite good insulator this sandwich has fairly complicated electric properties. In particular, at low frequencies the whole system has to be represented by a complex equivalent electric circuit model. 

Helfrich, W. Molecular theory of flow alignment of nematic liquid crystals. J. Chem. Phys. 50, 100-106 (1969)... 

A qualitative picture, Fig. 10.4, shows the distance dependencies of the orientational order parameter for homeotropically aligned nematic liquid crystal at the solid substrate. The problem is to explain such dependencies [6]. The influence of the surface on the orientational order parameter may be discussed in terms of the modified Landau-de Gennes phase transition theory. Consider a semi-infinite nematic of area A being in contact with a substrate at z = 0 and uniform in the x and y directions. When writing the free energy density a surface term -W8(z)S must be added to the standard expansion of the bulk free energy density ... 

The macroscopic theory of elasticity can explain why longitudinal ridges and troughs on the surface of a glass are conducive to the planar homogeneous alignment of nematic liquid crystals [23]. For simplicity, a sinusoidal shape is chosen for the cross-section of a surface relief with the wavevector q directed along x, see Fig. 10.21a ... 

Besides aligning liquid crystals, external electric fields can also change the orientational order and thus the electro-optical properties of liquid crystals. When the long molecular axis of a liquid crystal molecule, whose anisotropy of polarizability is positive, is parallel to the applied field, the potential of the molecule is low. Thus the applied field suppresses the thermal flue-mation and increases the order parameter. Now we discuss how the orientational order of a nematic liquid crystal changes with applied fields. Using the Landau-de Gennes theory, the free energy density of a liquid crystal in an electric field (when the liquid erystal director is parallel to the field) is [4]... 

Viscosity, especially rotational viscosity (yi), plays a crucial role in the LCD response time. The response time of a nematic LC device is linearly proportional to yi [45]. The rotational viscosity of an aligned liquid crystal depends on the detailed molecular constituents, structure, intermolecular association, and temperature. As the temperamre increases, viscosity decreases rapidly. Several theories, rigorous or semi-empitical, have been developed in an attempt to account for the origin of the LC viscosity [46,47]. However, owing to the complicated anisotropic attractive and steric repulsive interactions among LC molecules, these theoretical results are not completely satisfactory [48,49]. 

There is an abundance of molecular theories for the conditions of formation of liquid crystals [3,6,7]. Crudely speaking, they express the idea that it is "easier" to pack a volume densely with a collection of rigid rods in an aligned fashion than randomly. 

Liquid crystal theory involves explicit consideration of molecular alignment, and how this is affects or is affected by changes in temperature, flow conditions, concentration of solvents, etc. It seems clear that modifications in continum theory are needed to cope with complications encountered in liquid crystal polymers. As I see it, some of the fault is associated with simplistic assumptions about directional ordering employed in theories of nematics, so It seems worthwhile to review some of the thinking about such matters. 

Cellulose microfibrils are deposited by cellulose synthases into the cell wall in often strikingly regular patterns. Here we discuss several mechanisms that have been put forward to explain the alignment of cellulose microfibrils that gives rise to ordered cell wall textures the hypothesis that cortical microtubules align cellulose microfibrils during their deposition, the liquid crystal hypothesis in which cellulose microfibrils self-assemble into textures after their deposition, the templated incorporation hypothesis, and the geometrical theory in which the density of active cellulose synthase complexes inside the plasma membrane may dictate the architecture of the cell wall. 

Archer, L.A., Larson, R.G. (1995) A molecular theory of flow alignment and tumbling in sheared nematic liquid crystals. Journal of Chemical Physics, 103 (8), 3108-3111. 

---