Model flexoelectric

As we have seen above, in membranes (living and model) flexoelectricity provides a linear relation between membrane curvature and membrane polarization or transmembrane voltage and membrane bending stress. It is thus closely related to mechanosensitivity and mechanotransduction, basic features of living systems." ... 

A series of model nematic liquid crystals (among them oxadiazole derivatives) with transverse dipole moments were used to study the flexoelectric effect in guest-host mixtures with a commercial liquid crystal host <2005CM6354>. 

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. 

During recent decades the molecular theory of flexoelectricity in nematic liquid crystals was developed further by various authors. " In particular, explicit expressions for the flexocoefiicients were obtained using the molecular-field approximation taking into account both steric repulsion and attraction between the molecules of polar shape. The influence of dipole-dipole correlations and molecular flexibility was later considered. Recently flexoelectric coefficients have been calculated numerically using the mean-field theory based on a simple surface intermolecular interaction model. This approach allows us to take into consideration the real molecular shape and to evaluate the flexocoefiicients for mesogenic molecules of different structures including dimers with flexible spacers. 

General expressions for the flexocoefiicients of nematic liquid crystals have been obtained in terms of the direct correlation function using the powerful density functional approach. These expressions have been used to obtain some interesting numerical results using the Perkus-Yevic approximation for the pair correlation function. The results from the density functional theory have also been used in computer simulations of flexoelectricity using model bent-core molecules interacting via the Gay-Berne potential. Alternative general expressions for the flexocoefiicients have... 

Finally note that the flexoelectric effect is also important in the smectic phase although the corresponding molecular theory is at a rudimentary stage. Recently a molecular model for the conventional and the so-called discrete flexoelectric effect in tilted smectic phases has been proposed... 

The molecular-statistical theory of flexoelectricity, presented in the previous sections, does not allow us to establish a direct relation between the flexocoefficients and the details of a particular molecular structure (except for permanent electric and steric dipoles) because the theory is based on simple model interaction potentials. A different version of the mean-field theory, which takes into consideration the real molecular shape, has recently been proposed by Ferrarini et This approach is based on the... 

J. Stelzer, R. Berardi and C. Zannoni, Flexoelectric coefficients for model pear shaped molecules from Monte-Carlo simulations, Mol. Cryst. Liq. Cryst. A 352(1), 187-194, (2000). doi 10.1080/10587250008023176... 

A.V. Emelyanenko and M.A. Osipov, Theoretical model for the discrete flexoelectric effect and a description for the sequence of intermediate smectic phases with increasing periodicity, Phys. Rev. E 68(5), 051703/1-16, (2003). doi 10.1103/PhysRevE.68.051703... 

N.T. Kirkman, T. Stirner and W.E. Hagston, Continuum modelling of hybrid-aligned nematic liquid crystal cells optical response and flexoelectricity-induced voltage shift, Liq. Cryst. 30(9), 1115-1122, (2003). doi 10.1080/02678290310001594562... 

In the following sections of this chapter we will summarize the direct, as well as the converse, flexoelectric measurements in fluid and elastomeric (dry or swollen) bent-core nematic liquid crystals, and try to explain these observations using the structural model outlined above. 

The presence of clusters in BC nematics is now well established from various measmements. Recent studies " have in fact indicated a ferroelectric or an antiferroelectric response to an applied electric field, and an unusual low-frequency (presumably collective) mode has been detected in the dielectric spectra of bent-core nematics, which might also be related to clusters. In spite of the intense studies, however, the exact structure and the physical properties of the clusters are still unknown. Therefore, not surprisingly, a precise physical model for the role of polar clusters in the flexoelectric response of BC nematics and a quantitative estimation of the resulting increment of the flexocoefiicients has not yet been worked out. 

A.G. Petrov, Flexoelectric model for active transport, In ed. J. Vassileva, Physical and Chemical Bases of Biological Information Transfer, Plenum Press, New York, 1975. pp. 167. 

The chapter is organized as follows Section 4.2 describes flexodomains in the planar geometry. Particular emphasis is placed on the most recent theoretical results, where for the first time arbitrary ratios of the elastic constants ifi, if2 are considered as well as driving by an AC electric field. Section 4.3 deals with the effects of flexoelectricity on dissipative EC patterns. The focus is on qualitatively new phenomena that are not covered by the standard model of EC. In Section 4.4 we analyse the competition between flexodomains and EC patterns at low AC driving frequencies. The chapter ends with a discussion and some concluding remarks in Section 4.5. 

For the parameter combination o < 0 and <7a < 0, which can be found in some nematic compounds, electroconvection is definitely excluded within the standard model. Nevertheless, EC has surprisingly been observed in this case (for recent examples see, e.g. Kochowska et alP and Toth-Katona et al ). The theoretical analysis has proved that flexoelectricity is crucial for understanding this non-standard EC because in Eq. (4.7) the contribution V Pfi to Pel is dominant. It is interesting that the flexotorque on the director is determined by the difference (ei — 63) of the flexocoefScients while the sum (ei - - 63) governs the flexocharge and thus its contribution to the viscous torque. Further details will be sketched in Section 4.3.2. 

The chapter is organized as follows The second section discusses the prototype polar smectics the ferroelectric liquid crystals. We discuss the structure of the ferroelectric phase, the theoretical explanation for it and we introduce the flexoelectric effect in chiral polar smectics. Next we introduce a new set of chiral polar smectics, the antiferroelectric liquid crystals, and we describe the structures of different phases found in these systems. We present the discrete theoretical modelling approach, which experimentally consistently describes the phases and their properties. Then we introduce the discrete form of the flexoelectric effect in these systems and show that without flexoelectricity no interactions of longer range would be significant and therefore no structures with longer periods than two layers would be stable. We discuss also a few phenomena that are related to the complexity of the structures, such as the existence of a longitudinal, i.e. parallel to the... 

Which of the phases are important for flexoelectricity As we shall see below, the flexoelectric effect is the main reason for the large variety of phases. The flexoelectric interaction is actually the reason for significant interactions with the more distant layers. In addition, phases with larger phase differences are a source of another phenomenon the local polarization can also have a component parallel to the tilt direction of the polarization. However, to understand the richness of the phenomena, let us first focus on the phenomenological model, which describes all the phases above, their properties and the phase sequence. 

The theory and experiments of lyotropic and biomembrane flexoelectricity are reviewed. Flexoelectricity is a reciprocal relation between electricity and mechanics in soft lyotropic systems, i.e., between curvature and polarization. Experimental evidence of model and biomembrane flexoelectricity (including the direct and the converse flexoelectric effects) is reported. The biological implications of flexoelectricity are underlined. Flexoelectricity enables membrane structures to function like soft micromachines and nanomachines, sensors and actuators, thus providing important input to nanoionics apphcations. Nanobio examples include membrane transport, membrane contact, mechanosensitiv-ity, electromotility, hearing, nerve conduction, etc. 

Flexoelectricity is a basic mechano-electric effect that enables the nanometre-thick membranes of living matter to function like soft machines, thus converting the electrical stimuli of the hving world into mechanical ones, and vice versa. It also allows, by using model nanomembranes, the construction of mechanosensors and actuators for nanoionics applications. 

Furthermore, monolayer measurements demonstrate the variation of the dipole moment of peripheral proteins under stretching or compression. According to us, this implies the possibility of bimorph flexoelectricity of peripheral proteins (an analogue of the piezoelectricity of a bimorph plate), especially if these are symmetrically adsorbed over the two membrane interfaces, as suggested in the Danielli-Davson model. ... 

A model using the direct flexoelectric effect for the transformation of mechanical into electrical energy in the hearing process in stereocilia has been proposed." ... 

A.G. Petrov, Flexoelectricity and ion channels A confirmation of the flexoelectric model for ion transport. Cell. Molec. Biol. Lett. 2, suppl. 1, 231-253, (1997). 

L.M. Bhnov, M.l. Barnik, H. Ohoka, M. Ozaki and K. Yoshino, Separate measurements of the flexoelectric and surface polarization in a model nematic liquid crystal p-methoxybenzyhdene-p -butylanihne Validity of the quadrupolar approach, Phys. Rev. E 64(3), 031707/1-7, (2001). 

Blinov, L.M., Bamik, M.I., Ohoka, H., Ozaki, M., Yoshino, K. Separate Measurements of the Flexoelectric and Surface Polarization in a Model Nematic Liquid Crystal MBBA on Validity of the Quadrupolar Approach. Phys. Rev. E 64, 031707-031713 (2001)... 

In both the cases considered, an optical contrast of the patterns observed in isotropic liquids is very small. Certainly, the anisotropy of Uquid crystals brings new features in. For instance, the anisotropy of (helectric or diamagnetic susceptibility causes the Fredericks transition in nematics and wave like instabilities in cholesterics (see next Section), and the flexoelectric polarizaticm results in the field-controllable domain patterns. In turn, the anisotropy of electric conductivity is responsible for instability in the form of rolls to be discussed below. All these instabilities are not observed in the isotropic liquids and have an electric field threshold controlled by the corresponding parameters of anisotropy. In addition, due to the optical anisotropy, the contrast of the patterns that are driven by isotropic mechanisms , i.e. only indirectly dependent on anisotropy parameters, increases dramatically. Thanks to this, one can easily study specific features and mechanisms of different instability modes, both isotropic and anisotropic. The characteristic pattern formation is a special branch of physics dealing with a nonlinear response of dissipative media to external fields, and liquid crystals are suitable model objects for investigation of the relevant phenomena [39]. 

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