Charge flexoelectric

As possible explanations, several ideas have been proposed a hand-waving argument based on destabilization of twist fluctuations" [52], a possibility of an isotropic mechanism based on the non-uniform space charge distribution along the field [53] and the flexoelectric effect [55-57]. 

Fig. 3.13.2. Interpretation of the origin of flexoelectricity in an assembly of quadrupoles (a) in the undeformed state the symmetry is such that there is no bulk polarization, (6) a splay deformation causes the positive charges to approach the upper plane and to be pushed away from the lower one. This dissymmetry gives rise...

A. Todorov, A.G. Petrov and J.H. Fendler, Flexoelectricity of charged and dipolar bilayer Hpid membranes studied by stroboscopic interferometry, Langmuir 10(7), 2344 2350, (1994). doi 10.1021/la00019a053... 

According to Eq. (7.1) P is zero for the two cases of uniform director fields and pure twist. Hence both cases can serve as a zero state as far as flexoelectric excitations are concerned. It is important to note that a twist is not associated with a polarization (i.e. C2 is identically zero, cf. Fig. 7.2). An imstrained nematic has a centre of symmetry (centre of inversion). On the other hand, none of the elementary deformations - splay, twist or bend have a centre of symmetry. According to Curie s principle they could then be associated with the separation of charges analogous to the piezoeffect in solids. This is true for splay and bend but not for twist because of an additional symmetry in that case if we twist the adjacent directors in a nematic on either side of a reference point, there is always a two-fold symmetry axis along the director of the reference point. In fact, any axis perpendicular to the twist axis is such an axis. Due to this symmetry no vectorial property can exist perpendicular to the director. In other words, a twist does not lead to the separation of charges. This is the reason why twist states appear naturally in liquid crystals and are extremely common. It also means that an electric field cannot induce a twist just by itself in the bulk of a nematic. If anything it reduces the twist. A twist can only be induced in a situation where a field turns the director out of a direction that has previously been fixed by boundary conditions (which, for instance, happens in the pixels of an IPS display). 

The term piezoelectric was borrowed from the physics of solids by analogy to the piezoelectric effect in crystals without center of symmetry. As a rule, the piezoelectric polarization manifests itself as a charge on the surfaces of a crystal due to a translational deformation, e.g. compression or extension. Piezo-effects are also characteristic of polar liquid crystalline phases, e.g., of the chiral smectic C phase. The polarization, we are interested now, is caused by the mechanical curvature (or flexion) of the director field, and, following De Gennes, we call it flexoelectric. 

Fig. 11.25 Quadrupolar flexoelectric polarization. Undistorted nematic phase consisted solely of molecular quadrupoles (a) and appearance of a polar axis and flexoelectric polarization due to splay distortion (b). Note that in the lower part of (b) the density of positive charges is larger than in the upper part whereas in sketch (a) these densities are equal...

We measure pyroelectric coefficient y = dP/dT, using heating the hybrid cell by short ( 10 ns) laser pulses, as shown in Fig. 10.13. The only difference from the surface polarization measurements is using a hybrid cell instead of uniform (planar or homeotropic) cells [28]. The laser pulse produces a temperature increment about AT 0.05 K and the flexoelectric polarization changes. To compensate this change, a charge passes through the external circuit and the current i = dqldt is measured by an oscilloscope. From the identity (A is cell area)... 

The characteristic dependences A oc and the absence of the threshold were determined experimentally [185], and the coefficient e33 = 3.7 x 10 dyn / was determined for MBBA. The sign of the flexoelectric coefficient was determined in a separate experiment, where the velocity of motion of the liquid crystal due to the force QE was recorded optically. It is essential that the space charge Q occurs as a result of the flexoelectric effect Q = —dP/dy. For MBBA, 633 was found to be positive. 

The most direct method of finding the coefficient en would be to fill the space between the metallic coaxial cylinders with a liquid crystal having pear-shaped molecules, with the surfaces of the cylinders having been pretreated for homeotropic orientation and to measure the potential difference between the cylinders. In fact, because of the difference in radii of the cylinders, the nematic liquid crystal structure proves to be splay deformed, and if the molecules have even a small longitudinal dipole moment the plates of the coaxial capacitor would be charged. However, despite its apparent simplicity, this experiment is, in fact, complicated because of the screening of the potential caused by the flexoelectric effect by firee charges from the liquid crystal and the atmosphere. 

Figure 33. The symmetry of twist. If we locally twist the adjacent directors in a nematic on either side of a reference point, there is always a twofold symmetry axis along the director of the reference point. Therefore a twist deformation cannot lead to the separation of charges. Thus a nematic has only two nonzero flexoelectric coefficients.

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