Director curvature, components

Flexoelectric polarization of FLC layers should also be taken into account [4, 29]. Reference [29] shows that the total polarization P includes two components, P and P , which are parallel and perpendicular to the C2 axis, respectively. The components P and partially f , are induced by the director curvature in the layers or by the flexoelectric effect (Fig. 7.4)... 

In this coordinate system the director curvature can be separated into six components ... 

We neglect higher order terms in the expansion as we are concerned only with infinitesimal deformations. Since these deformations relate to changes in the orientation of the director, may appropriately be called the curvature strain tensor (Frank ). In order to define the components of this tensor, let us choose a local right-handed system of cartesian coordinates with n parallel to z at the origin. The components of strain are then... 

For a single, two-dimensional liquid crystal hpid bilayer, or a membrane, in a smectic A-like state, the director field is represented by the membrane normal n. Fiexoelectricity is then defined as a curvature-induced area membrane polarization, or, conversely, as an electric field-induced membrane curvature. Lipids and proteins are oriented parallel to each other along the local membrane normal in the flat state. A curvature of the membrane surface leads, indeed, to a splay type deformation of the molecular local director, with a splay vector S = (si - - S2)n, while a bend deformation along the membrane normal is not allowed because there is no third dimension. Then, obviously, the only polarization component points along the membrane normal. 

This relation expresses the fact that the elastic energy is a quadratic form in three curvature deformations or strains, now called splay, twist, and bend, which we can treat as independent. They are sketched in Fig. 31. The splay is described by a scalar (a pure divergence), V-n, the twist is described by a pseudoscalar (it changes sign on reflection in a plane parallel to the twist axis or when we go from a right-handed to a left-handed reference frame), which is the component of Vxn along the director, (F x/i(n, whereas the bend is described by a vector with the component of Vxn perpendicular to the director, f x x. 

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