Determination of the flexoelectric coefficients

If the molecule possesses shape polarity in addition to a permanent electric dipole moment then the possibility exists that a splay or bend deformation 

1 Meyer s model of curvature electricity. The nematic medium composed of polar molecules is non-polar in the undeformed state ((a) and ( )) but polar under splay (b) or bend ( /). (After Meyer. ) 

In a first order theory, the polarization P should be proportional to the distortion  

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P.R. Maheswara Murthy, V.A. Raghunathan and N.V. Madhusudana, Experimental determination of the flexoelectric coefficients of some nematic liquid crystals, Liq. Cryst. 14(2), 483-496, (1993). 

Fig. 3.13.3. A hybrid aligned cell for the determination of the anisotropy of the flexoelectric coefficients. In this geometry, the director has a splay-bend distortion which gives rise to a flexoelectric polarization P. On applying an electric field E, the director is twisted by an angle (j> cc — which can be measured optically.

The general mean-field results, presented in this section, enable us to clarify this problem. It should be noted that Straley s theory was developed for a system of rigid rods and thus it takes into consideration only a short-range steric repulsion between molecules. On the other hand, in the theory of Helfrich and Petrov and Derzhanski the flexocoefficients are expressed in terms of Frank elastic constants, which, in turn, are determined by both the intermolecular attraction and repulsion. The relation between the two contributions can be clarified using Eqs (1.31) and (1.32), which can be used to obtain the following estimate of the flexoelectric coefficients ... 

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. 

Flexoelectric coefficients are mainly determined by the function 12, which specifies the molecular shape, and the pair attraction interaction potential V(l,2). In the general case neither of these functions can be written in a simple analytical form, which makes it very difficult to evaluate the flexoelectric coefficients using the general formulae. At the same time it... 

Secondly, it follows from Eqs (1.31) and (1.32) that the longitudinal molecular dipole d provides a much smaller contribution to the flexocoef-ficients than the transverse dipole d , since A/k 10. Thus we conclude that the dipole flexoeffect is expected to be important only for molecules with large transverse dipoles. Note that the significant dipole flexoeffect has indeed been determined for nematics composed of molecules with large transverse dipoles. For cyanobiphenyl liquid crystals Marcerou and Frost did not find any dipolar flexoelectric effect, which may be determined not only by the tendency to form dimers with antiparallel dipoles but also by a relatively small contribution from transverse molecular dipoles to the flexoelectric coefficients. 

E.K. Tidey, L.A. Parry-Jones and S.J. Elston, Determination of the difference of flexoelectric coefficients in a nematic liquid crystal using a conoscopic technique, Liq. Cryst. 34(2), 251-255, (2007). 

The molecular statistical approach to calculate the flexoelectric coefficients was developed independently by Helfrich and Derzhanski and Petrov. The calculation is based on the requirement to ensure maximum packing condition. The excess number (AN = N+ - N-) of the molecules with dipole moment n determines the electric polarization P= AN/i. Dividing this by the distortion we get the flexoelectric coefficient. 

As mentioned above, the dipolar flexoelectric coefficients are determined by the polar molecular shape, which can be characterized by the molecular steric dipole. For a molecule having the shape of a truncated cone, as shown in Fig. 1.2, the steric dipole is in the direction of the long molecular axis a and is proportional to the cone angle 7, while for a bent-rod molecule the steric dipole is parallel to the short axis b and is proportional to the bend angle 7x- The relation between the flexocoefficients and the molecular shape is determined by the distance of closest approach 12 = i2(xi,X2,ri2), which reflects the polarity of the shape. 

Table 6.1. Flexoelectric coefficients of bilayer lipid membranes made from different lipids under various ionic conditions. 1-3 electrical estimation of curvature 4-7 interferometric mesisurement of curvature, sign determination of / not attempted. All experiments except 7 concern the direct fiexoeffect. All data refer to the high-frequency range above 200-300 Hz, i.e., to the blocked flexocoefficient (from Petrov, with kind permission from Taylor and FVancis Group).

Today the electrooptical properties of liquid crystals form well-developed branches both in the physics and technology of liquid crystals. In addition, electrooptical measurements are the basis of a number of precise methods for determining the physical parameters of a material, such as its elastic and viscosity coefficients, optical anisotropy, spontaneous polarization, flexoelectric coefficients, anchoring energies at interfaces, etc. 

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