The order parameter dependence of the flexoelectric coefficients in nematic liquid crystals

2014 Within the general framework of Landau-de Gennes theory for nematic liquid crystals it is shown that the difference of flexoelectric coefficients e* = e1 e3 ~ S2, where S is the nematic order parameter. This result agrees with the molecular theory but is in contradiction with recent measurements of the constant e* which yield e* ~ S. The contradiction can be resolved by the inclusion of conformational freedom into the molecular theory. J. Physique Lett. 45 (1984) L-823 L-826 15 AOUT 1984, 1 Classification Physics Abstracts 61.30 The flexoelectric effect in liquid crystals manifests itself in the appearance of induced polarization under the condition of a curvature strain. In the nematic phase there are two independent contributions to the induced polarization P : where n is the director and el, e3 are the flexocoefficients. Two different microscopic interpretations of the flexoelectric effect have been suggested in the literature. The first one was proposed by Meyer [1] who assumed that macroscopic polarization is related to the orientation of asymmetric dipolar molecules in the distorted liquid crystal. The corresponding molecular theory has been developed in [2-5]. Prost and Marcerou [6] have taken into account the fact that macroscopic polarization can be caused by the gradient of the quadrupole density. According to Marcerou and Prost [7], the measurement of the temperature dependence of flexocoefficients can help to distinguish between these two interpretations. Indeed, the dipolar contribution is proportional to S 2 [1,2] while the quadrupolar contribution is proportional to S [6], where S is the nematic order parameter. In a recent study [5], however, we have shown that a consistent molecular-statistical theory of the flexoelectric effect results in both S and S 2 dependence of flexocoefficients (1 ). At (1) This result was first obtained by J. P. Straley [4] for the gas of long rods. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019840045016082300 L-824 JOURNAL DE PHYSIQUE LETTRES the same time, all molecular theories of the flexoelectric effect [2-6] (both dipolar and quadrupolar) yield the S 2 dependence of the difference offlexocoefficients e* = el e3. Recently Dozov, Martinot-Lagarde and Durand [8] have described the first direct measurement of the flexoelectric constant e* and have concluded that e* possesses the unexpected S-like temperature dependence in the nematic compound octyloxy cyanobiphenyl (80CB). This result is in contradiction with the existing molecular theories [2-5]. However, the molecular theory is based on various simplifications and assumptions and thus it can yield the incorrect order parameter dependence of flexocoefficients if, for example, the inadequate molecular model is used in the calculations. According to [8], the S-like temperature dependence of the constant e* is attributed to the conformational freedom of the long alkoxy chain of the 80CB molecule. The molecular flexibility was not taken into account in the molecular theory [2-5]. In the present paper it will be shown that the relation e* ’"’-I S 2 is the general result of the phenomenological Landau-de Gennes theory for nematic liquid crystals and hence this result is not related to the shortcomings of molecular theory. Therefore we arrive at the more serious contradiction between the theory and the direct experiment. This contradiction indicates that the order parameter expansion of the free energy cannot be always used in the description of the nematic phase. In conclusion we discuss the relation between the molecular flexibility and the order parameter dependence of the .flexoelectric constant In Meyer’s continuum,, theory [1] the contribution to the free energy of the nematic phase, which describes the flexoelectric effect, can be written in the form where E is the external electric field, X is the dielectric susceptibility tensor, xxa = xl 6.p + AX n,, np. Minimizing the free energy (2) with respect to the polarization P, one obtains expression (1) for P with e1 = ei /)), and e3 = e3 xl. Note that the constants el and e3 are temperature dependent. On the other hand, the free energy of the nematic liquid crystal can be expanded in terms of the nematic order parameter Qx~(r) = S(n~(r) n~(r) 2013 (1/3) ~), a, ~i = x, y, z. In this case the free energy 6FFE contains all first-order gradient terms which are linear in P. Taking into account only the terms which are first and second-order in S, one obtains