The Flexoelectric Coefficients

BLM, the type and concentrations) of the electrolytes bathing the BLM, the ions adsorbed on the BLM surface, and the extent to and frequency with which the BLM is bent [419]. These experimental observations have led to a phenomenological definition of the flexoelectric coefficient, f, as the ratio between the bending-induced transmembrane potential, Uf, and the change of curvature, c, that accompanies the bending of the membrane ... 

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.

In this book the flexoelectric effect is mainly considered from the phenomenological point of view. At the same time it is very interesting and important to reveal the molecular origin of flexoelectricity and, in particular, to consider different types of intermolecular interactions that may be responsible for the dipolar ordering in a deformed liquid crystal, and to study the effects of intermolecular correlations and the molecular structure. This problem can only be solved using a molecular-statistical theory, which eventually allows us to express the flexoelectric coefficients in terms of molecular model parameters using various approximations. 

Thus the quadrupole mechanism yields very simple expressions for the flexo-electric coefficients, which are proportional to the nematic order parameter S in the first approximation (usually the parameter D is much smaller than S). In addition, the difference of the flexoelectric coefficients appears to be equal to zero, i.e. Ae = ei — 03 = 0 if the quadrupole contribution alone is taken into account. 

Simple expressions for the flexoelectric coefficients in the nematic phase can be obtained using the generalized van der Waals approximation. In this approximation the intermolecular attraction is taken into account in the mean-field approximation while the steric repulsion is accounted for by taking into consideration short-range steric correlations via the excluded... 

In the present section we consider only the flexoelectric effect relating to the polar molecular shape, and we therefore assume that the pair attraction interaction potential V(xi,X2,ri2) is even in ai,a2,bi,b2. In this case the first term in Eq. (1.30) does not contribute to the flexoelectric coefficients, which are determined mainly by steric dipoles. 

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... 

It follows from Eqs (1.31) and (1.32) that the predominant contribution to the flexoelectric coefficients is determined by the isotropic intermolecular attraction modulated by the polar molecular shape. Indeed, in the general case the maximum attraction interaction energy V(R) kT where R is the equilibrium distance between the two molecules. It follows then that A ... 

As mentioned above, the first expressions for the flexoelectric coefficients were obtained by Helfrich and Petrov and Derzhanski while a systematic molecular-statistical theory was developed later by Straley. The results of these two approaches were compared by Marcerou and Prost who concluded that the theories of Helfrich and Petrov and Derzhanski and of Straley describe different mechanisms for the dipolar flexoelectric effect because Straley s theory 5nelds values for the flexocoefficients that are two orders of magnitude smaller than the experimental ones, and which therefore can be neglected. 

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 ... 

Let us now discuss the approximate expressions for the flexoelectric coefficients, Eq. (1.31), in more detail. Firstly, note that the expressions for both coefficients and es contain terms proportional to both S and S. It has been assumed in the literature that the dipolar contribution to the flexoelectric coefficients is always proportional to while the quadrupole contribution is proportional to S, and even the method of separation between the dipolar and quadrupolar flexoelectric effect has been proposed based on these preliminary results. The results of the consistent molecular theory presented in this section allow us to conclude that the relation e S for the dipolar contribution is due to the shortcomings of the semi-phenomenological approach. The results of this section also cast some doubt on the quantitative ratio of the dipolar and quadrupole contributions based on a comparison of the two terms in the expression e = eoS + C2S. At the same time, the absence of the linear term in S in the dependence e S) for a number of nematic materials stUl points to the predominant role of the quadrupole flexoeffect for those materials. 

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. 

The expressions for the flexoelectric coefficients presented in this section are derived using the molecular-field approximation. Therefore, care should be taken in the description of nematic liquid crystals composed of strongly polar molecules. In such liquid crystal materials (for example, cyanobiphenyls) the flexoelectric coefficients may be strongly affected by the short-range dipole-dipole correlations, which are considered in the following section. 

The general expressions for the flexoelectric coefficients obtained using the density functional approach can be used to estimate the contributions from different types of intermolecular correlations, including short-range dipole-dipole correlations. We use the following approximation for the direct pair correlation function ... 

The linear terms in the expansion Eq. (1.39) do not contribute to the flexoelectric coefficients because the dipole-dipole interaction potential is odd both in di and d2 and hence the corresponding contributions vanish after averaging over the orientation of the molecular axes. Thus it is necessary to take into account the quadratic terms in the expansion of the direct correlation function. Then the contribution from the dipole-dipole correlations to the flexocoefficients can be written in the form ... 

Substituting Eqs (1.42) and (1.43) into Eq. (1.44) and then into the general Eq. (1.40) we obtain the following expressions for the contributions from the dipole-dipole correlations to the flexoelectric coefficients of the nematic phase °... 

It follows from Eq. (1.45) that the contribution from the dipole-dipole correlations strongly depends on the value of the molecular dipoles, i.e. A dj (i if d > d . Therefore the correlation contribution can be substantial in liquid crystal materials composed of strongly polar molecules. The effect of dipole-dipole correlations on the flexoelectric coefficients has been discussed in experimental papers. 

It should be noted that dipole-dipole correlations may contribute to the flexoelectric coefficients only when mesogenic molecules have both a longitudinal and a sufficiently large transverse dipole. This may explain why the correlation contribution seems to be very important for oxycyanobiphenyls (and not for cyanobiphenyls, which do not possess any transverse dipoles). For weakly polar molecules d dx 1 D, and in this case the contribution from the dipole-dipole correlations is two orders of magnitude smaller than for 80CB and can be neglected. 

M.A. Osipov, The order parameter dependence of the flexoelectric coefficients in nematic liquid crystals, J. Physique Lett. 45(16), 823-826, (1984). 

A. Ferrarini, C. Greco and G.R. Luckhurst, On the flexoelectric coefficients of liquid crystal monomers and dimers A computational methodology bridging length-scales, J. Mater. Chem. 17(11), 1039-1042, (2007). 

H.J. Deuling, On a method to measure the flexoelectric coefficients of nematic liquid crystals. Solid State Commun. 14(11), 1073-1074, (1974). 

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. 3.10. Magnitude of the flexoelectric coefficient e3 versus the temperature deviation from the nematic-isotropic phase transition for (a) ClPbislOBB and for (b) 5CB. Data taken from Harden et al Note the three orders of magnitude difference in the vertical scales ...

Fig. 3.13. Relative temperature dependences for the BC nematic lODClPBCP fluid monomer, for a BC nematic swollen in a calamitic liquid crystal elastomer (BCN-LCE) and a BC nematic elastomer (BCLCE). The flexoelectric coefficient of an LCE is also shown (note that it is not distinguishable from the horizontal axis at the present scale due to the three orders of magnitude difference).

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