Director gradient

The vector h is the so-called molecular field, it is produced by the same director gradients that produce the distortional energy Wd- Specifically,... 

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]... 

Since director gradients are neglected, the elastic part of the stress tensor = 0. Thus, for n = (1,0,0) we have from (3.3.5)... 

An alternative interpretation has been proposed by Barbero and co-workers [195, 218]. These authors expand the free energy to fourth order in the director derivatives. This increases the number of elastic terms by 35 and makes a general application of this so-called second-order elastic theory rather complicated. While this approach provides the correct torque balances, it results in large subsurface director deformations on a molecular length scale which are not consistent with the ideas of Frank s theory, which is based on the assumption of weak director gradients. 

As far as the director gradients are concerned we have chosen to use the form... 

The Ericksen-Leslie theory from Section 4.2.5 will be used with all director gradients and the elastic energy being set to zero, so that we are dealing with an anisotropic fluid. Incorporating the gravitational potential the relevant dynamic equations... 

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