Director gradients, nematics

Because nematic liquid-crystalline polymers by definition are both anisotropic and polymeric, they show elastic effects of at least two different kinds. They have director gradient elasticity because they are nematic, and they have molecular elasticity because they are polymeric. As discussed in Section 10.2.2, Frank gradient elastic forces are produced when flow creates inhomogeneities or gradients in the continuum director field. Molecular elasticity, on the other hand, is generated when the flow is strong enough to shift the molecular order parameter S = S2 from its equilibrium value 5 . (Microcrystallites, if present, can produce a third type of elasticity see Section 11.3.6.)... 

The thermodynamical equilibrium of nematics would correspond to a spatially uniform (constant n(r)) director orientation. External influences, like boundaries or external fields, often lead to spatial distortions of the director field. This results in an elastic increment, fd, of the volume/ree energy density which is quadratic in the director gradients [2, 3] ... 

Though nematics are non-polar substances, a polarization may emerge in the presence of director gradients, even in the absence of an electric field. This flexoelectric polarization [2, 3]... 

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

Ratio of the shear stress, a, to the shear velocity gradient, y, for a nematic liquid crystal with a particular director orientation, denoted by /, under the action of an external field ... 

Note 1 The three Miesowicz coefficients (//i, 772, and 773) describe the shear flow of a nematic phase with three different director orientations, (see Fig. 31) namely 771 for the director parallel to the shear-flow axis 772 for the director parallel to the velocity gradient and 773 for the director perpendicular to the shear flow and to the velocity gradient. 

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). 

The magnitudes of the viscosities (the a ,- s) for a single small-molecule nematic can differ from one another by an order of magnitude or more. As a result, the fluid s resistance to flow depends strongly on the directions of the flow and the flow gradient relative to the nematic director. In a shearing flow, the viscosities o 2 and o 3 determine director torques in the orientations shown in Fig. 10-7b and 10-7c. If the director is oriented in the flow direction... 

Consider now the steady laminar flow of a nematic fluid between two parallel plates. If the flow is along x and the velocity gradient along y the components of the velocity and the director are... 

Fig. 3.6.9. (a) Ratio of transverse to longitudinal pressure gradient versus director orientation in plane Poiseuille flow of nematic MBBA. Sample thickness d = 200/m. Length of cell L = 4cm and lateral width 1=4cm. (b) E> ection f of flow lines with respect to the y axis versus in wide cells (L/l ). Full line represents theoretical variation. (From Pieranski and Guyon. )... 

Order electricity may be expected to manifest itself at the nematic-isotropic (or air) interface where, as discussed in 2.7, the order parameter changes rapidly across the transition zone from one phase to the other. Let us make the simple assumption that at the N-I interface the gradient of the order parameter sj, where is a coherence length. If is the component of the polarization normal to the interface created by the order parameter gradient, and the director at the interface is tilted at an angle 6 with respect to z, the dielectric energy due to order electricity will be proportional to ... 

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