Mechanism flexoelectric

Flexoelectricity involves two degrees of freedom of the BLM electrical and mechanical. The system is amenable to simultaneous electrical and optical investigation of mechanical-to-electrical-energy conversion mediated by a bi-molecularly thin membrane. BLMs themselves are unlikely to be used as device components. They offer, however, an eminently suitable means for conducting the fundamental studies which are necessary for the full potential of the membrane-mimetic approach to advanced materials to be realized. 

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]. 

Figure 6.26 The flexoelectric effect in BaTiO (a) the evolution of the potential energy curve under a homogeneous stress and in a strain gradient (b) domain switching via mechanical stress imposed by a probe (Original data from Lu et al. (2012))...

It should be noted, however, that the flexoelectric effect is not necessarily related to the ordering of molecular dipoles. Frost and Marcerou proposed another microscopic mechanism of the flexoelectric effect, which requires neither the asymmetry of the molecular shape nor the permanent molecular dipole. The macroscopic polarization may simply appear in the direction of the gradient of average density of the molecular quadrupole moments. The quadrupole mechanism of flexoelectricity is more general because, in principle, it should manifest itself in any anisotropic material with a non-zero quadrupole density including solid crystals d and elastomers. 

This chapter is arranged as follows. In Section 1.2 we consider in more detail the dipolar and quadrupolar mechanisms of flexoelectricity, and in Section 1.3 we derive the general expressions for the flexocoefficients in terms of the direct pair correlation function. These results are used in Section 1.4 to obtain approximate expressions for the flexocoefficients in the molecular-field approximation taking into account both intermolecular repulsion and attraction. In that section we also consider the dependence of the flexocoefficients on the absolute value of the molecular dipole and on the orientation of the electric dipole with respect to the molecular long axes and the steric dipole. In Section 1.5 the effect of dipole-dipole correlations is analysed and in Section 1.6 we discuss the mean-field theory of flexoelectricity, which allows us to account for the real molecular shape. 

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. 

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. 

Since the flexoelectric effect is associated with curvature distortions of the director field it seems natural to expect that the splay and bend elastic constants themselves may have contributions from flexoelectricity. The shape polarity of the molecules invoked by Meyer will have a direct mechanical influence independently of flexoelectricity and can be expected to lower the relevant elastic constants.The flexoelectric polarization will generate an electrostatic self-energy and hence make an independent contribution to the elastic constants. In the absence of any external field, the electric displacement D = 0 and the flexoelectric polarization generates an internal field E = —P/eo, where eq is the vacuum dielectric constant. Considering only a director deformation confined to a plane, and described by a polar angle 9 z), and in the absence of ionic screening, the energy density due to a splay-bend deformation reads as ... 

The magnitude of the effect is characterized by two flexoelectric coefficients, ei and 63, for splay and bend, respectively. As far as the microscopic origin of these phenomenological flexocoefficients is concerned, two different mechanisms - dipolar and quadrupolar - have been identified they are discussed in detail in Chapter 1. ... 

Converse flexoelectric studies of lyotropic liquid crystals, such as vesicles, is still an active subject. Notably, the sensory mechanism of outer hair cell composite membranes " can be understood by the flexoelectric properties of the lipid bilayer. The converse of this effect, i.e., a voltage-generated curvature, has also been observed and was discussed by Todorov et Another related phenomenon is the ferroelectricity which results from the tilted layered structures of chiral molecules, which has been discussed extensively since the 1980s.Ferroelectric phases are called... 

Summarizing, experimental observations suggest that the giant (direct or converse) flexoelectricity of bent-core nematics is related to the polar smectic clusters occurring in them. In order to explore the exact mechanism for how clusters contribute to the flexoelectric response, further experimental and theoretical studies are needed. 

Patterns, i.e. regular spatiotemporal structures, can easily be generated in liquid crystals via a large variety of external stresses, e.g., by mechanical shear, temperature or pressure gradients, electric or magnetic fields, etc. representative examples can be found in Buka and Kramer. Here we concentrate on patterns induced by electric fields in nematics and in particular on the implications of flexoelectricity. 

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. 

Phenomenologically, there is a close analogy between flexoelectricity of thermotropics and lyotropics. Equations (6.1) and (6.2) are in correspondence since a bend deformation of the director is not allowed either in a two-dimensional bilayer or in a lamellar lyotropic phase. From dimensional arguments we have concluded that the three-dimensional to two-dimensional correspondence leads to / = e d, where d is the bilayer thickness. However, the molecular mechanisms of the two are very different, e.g., molecular shape asymmetry is not a precondition for dipolar membrane flexoelectricity (see Petrov and below). 

Exploring the molecular mechanisms of flexoelectricity is a central task of the liquid crystal approach in membrane biophysics. 43,15-21 pjjg flexoelectric coefficient can be represented as an integral over the curvature derivative of the distribution of the normal component of polarization P z, c+) across the membrane (c+ = ci - - C2 is the total curvature). Both direct and converse flexocoefficients can be expressed in this manner and these can be shown to be equal ... 

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

The elemental motile unit of an OHC strikingly resembles the flexoelectric domain structure whose calculated period 7rifn(ei ) also depends linearly on the inverse electric field (Fig. 6.9). Such a repetitive arc and pillar nano-architecture, containing sharp points at the confluence of any two adjacent arcs, is inherently polar and enhances the flexoelectric mechanism (e.g., a sinusoidal-shaped membrane will not enhance flexoelectricity while one of the half-waves is reduced the opposite one will be extended and vice versa). This arc motif is repeated a few thousand times along the... 

Mechanical tension may be due to a converse-flexoelectric area dilatation induced by the photopotential, as analysed above [c/. Eq. (6.16)]. The photopotential is due to the pH gradient because of the ferricyanide photochemical reactions. The increase of membrane tension above a certain threshold can open the ion channels thus dissipating the pH gradient returning the membrane tension back to zero (at this instant extensive form fluctuations, cf. Fig. 6.10, are thermally excited), so that the channels close and the cycle repeats. Under this scenario it is clear that the whole photo-... 

Flexoelectricity is a reciprocal relation between the electrical and mechanical properties of liquid crystalline biostructures. We have underlined in... 

A.G. Petrov, Electricity and mechanics of biomembrane systems Flexoelectricity in living membranes, Anal. Chim. Acta 568(1), 70-83, (2006). 

W.E. BrowneU, Membrane based motor mechanisms. 1st World Flexoelectric Congress, SUNY-Buffalo, 2001. 

L.A. Parry-Jones, R.B. Meyer and S.J. Elston, Mechanisms of flexoelectric switching in a zenithally bistable nematic device, J. Appl. Phys. 106(1),... 

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