Interaction flexoelectric

Second, for observing curvature structures induced by an externally applied electric field, one must also consider the interaction of the field with the anisotropy of the polarizability of the medium, which tends to align the director either parallel or perpendicular to the applied field. This can suppress the flexoelectrically induced curvature in many geometries. 

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... 

The results of Straley can be obtained by neglecting the pair attraction interaction potential V(xi, X2, ri2) in the equation for the direct correlation function. Indeed, the Straley theory of flexoelectricity was developed for the system of hard polar rods, while for thermotropic liquid crystals both the molecular shape and the intermolecular attraction are important. 

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 ... 

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 ... 

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... 

We conclude that the existing molecular theory of flexoelectricity is far from being complete. Any future more advanced theory should combine a more sophisticated molecular-statistical approach, which will allow us to take into account at least short-range intermolecular correlations, and more realistic intermolecular interaction potentials, which reflect the real molecular structure. 

This discrepancy clearly indicates that the BC nematics characterized by the presence of polar clusters cannot be regarded - at least from the point of view of the flexoelectric response - as a homogeneous continuum instead the behaviour of the clusters in and their interaction with the surrounding nematic matrix should be handled separately. This challenging problem still waits for a solution. 

In this chapter we consider several important aspects of the flexoelectric effect for chiral polar smectic liquid crystals and for the variety of phases. First, we discuss the reason for indirect interlayer interactions, which extend to more distant layers, and the lock-in to multi-layer structures. Second, although it was believed for a long time that polarization in tilted chiral smectics is always perpendicular to the tilt with the smectic layer normal, a component in the direction of the tilt may exist. And third, in multi-layer structures, the flexoelectricaUy induced polarization can be extremely large but is difficult to measure. 

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... 

In the previous section we described terms that contribute to the free energy of the system, however, without considering interactions where polarization is involved. In this section we will take a closer look at interactions that appear due to the polarization or influence polarization, especially flexoelectricity. 

We have seen that fiexoelectricity (and piezoelectricity) influences the material properties expressed in the coefficients of Eq. (5.16) and consequently influences quantitatively the pitch, the phase transition temperature and the tilt. But there are no qualitative changes, i.e. no new stable phases with qualitatively different structures appear due to these two interactions. Below we shall see that in systems where the interactions are more complex the role of the flexoelectric phenomenon can be very significant and it is crucial for the stability of many structures. 

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 expression in Eq. (5.19) is not limited to small changes from layer to layer. Let us now write the free energy in a discrete form. We will intentionally leave a discrete description for flexoelectric interactions for more detailed discussion later. Here we give the form of the free energy and in a discrete system show how to search for structures that minimize the free energy and how to verify their stabilities. We discuss also questions that remain open if flexoelectricity is not considered. The free energy is now the sum of the energies specific for each (jth) layer... 

In the previous section flexoelectric interactions were not considered in the free energy. We have also seen that only three of the structures found in antiferroelectric liquid crystals can be explained with the form of the free energy presented in the previous section. Let us first consider the discrete form of the flexoelectric effect and its influences on the theoretical description of the structures. We shall see that the flexoelectric effect is a source of indirect interactions between more distant layers and consequently the reason for all structures that cannot be expressed by the single phase difference. 

Flexoelectricity appears because interactions with molecules above and below the interacting layer are different. These interactions hinder rotation of the molecules in interacting layers in a different way, which affects the direction of the most favourable molecular orientation around the long molecular axis. As a consequence the direction and the magnitude of the polarization change. The flexoelectric interaction is described by the following contribution to the free energy... 

The historical theoretical explanation of the structures found in antifer-roelectric liquid crystals was actually different. Initially it was believed that anticlinic tilts in next nearest layers, which are the source of the competing interactions, are present because polar interactions with next nearest layers are strong enough. It was shown later that the flexoelectric effect is much more important and that these interactions are most probably the main source of the competition. 

The term is the result of the electrostatic free energy of the flexoelectrically induced part of the polarization. It can clearly be seen that this interaction originates in the flexoelectric effect and is always positive. The next nearest layers therefore favour an anticlinic orientation. The basic period of the structure, if this is the only interaction, would consist of four layers (Fig. 5.6, second row). In addition, if the flexoelectrically induced polarization is comparable to the piezoelectrically induced polarization, the indirect interaction with the next nearest layers can be surprisingly strong. 

In the expression Eq. (5.33) we can see that the flexoelectric effect when it is combined with the piezoelectric effect (the second part of the coefficient /i) has similar effects as direct chiral interactions due to the van der Waals field having chiral s mimetry around the chiral molecules given by /i. We cannot distinguish between the two components as the piezoelectric coefficient Cp and the coefficient describing direct chiral interactions /i probably depend equally (they are proportional) on the enantiomeric excess. 

Its origin is in the combination of the chiral piezo-interaction and the achiral flexoelectric interaction. It favours a perpendicular orientation in interacting layers, which means a period of eight layers (Fig. 5.6, fifth row). 

All consequences of the fiexoelectric effect considered so far are indirect and are not directly observable. Fiexoelectric interactions change the elastic constant (Eq. 5.16) in the SmC phase, which influences the layer polarization (Eq. 5.42). The layer polarization influences intralayer interactions and interactions to nearest layers and is consequently a source of interactions to more distant layers. The fiexoelectric interactions stabilize different structures with longer periodicities. But these are all indirect effects, which often cannot be separated from other effects. As the fiexoelectric effect is of achiral origin, there is no simple way to isolate the influence of flexoelectricity on the structure or the macroscopic properties. 

So, how can the flexoelectrically induced polarization be measured It would be natural to study interactions with external electric fields. However, it is not easy to observe the influences of the field on the structme, to measure the static or dynamic polarization in the electric field and similar. 

The fiexoelectric contribution to the layer polarization Pj is given in Eq. (5.42). For the interaction of a sample with an electric field due to the flexoelectrically induced polarization, we have to sum the contributions of all layers ... 

In more complex chiral polar smectics, antiferroelectric liquid crystals, there are many consequences of the flexoelectric effect. It influences interlayer interactions and causes indirect interactions between more distant layers to appear and become important. The phenomenon is the reason for the appearance of commensurate structures that extend up to six layers. In addition, longitudinal polarization, i.e. the polarization that has a component parallel to the tilt, exists in more complex structures such as the SmCpi2, the SmC jj and the SmC phases. Unfortunately it seems that flexoelectric polarization cannot be detected separately from other phenomena by simple means. A way of measuring the flexoelectric contribution in tilted polar smectics still seems to be an open question. 

I. Bivas and A.G. Petrov, Flexoelectric and steric interactions between two bilayer lipid membranes resulting from their curvature fluctuations, J. Theor. Biol. 88(3), 459-483, (1981). doi 10.1016/0022-5193(81)90277-0... 

Despite all these problems, flexoelectricity has, from time to time, been found wanting when the results of an experiment could not fully be interpreted by the dielectric interaction alone though its relevance or responsibility could not always be proved. 

In liquid crystals with the capability of flexoelectric effect, in the absence of external electric fields, the state with uniform director configuration, which has no induced polarization, is the ground state and is stable. When an electric field is applied to the liquid crystal, the uniform orientation becomes unstable, because any small orientation deformation produced by thermal fluctuation or boundary condition will induce a polarization which will interact with the electric field and results in a lower free energy. The torque on the molecules due to the applied field and... 

If Ae were negative but not very small, the dielectric interaction would prevent the deformation of the director configuration. We estimate the dielectric anisotropy Ae, which will make the flexoelectric effect disappear, in the following way. The dielectric energy is... 

The second turn of the discussion around the nature of a SHG in nematic liquid crystals arised when the SHG was observed in oriented layers of 4-methoxybenzylidene-4 -butylaniline (MBBA)/ The phenomenon has been explained by the lack of the symmetry center in the nematic phase. The zero-field SHG in MBBA was also investigated but the nature of the effect was connected with the flexoelectric polarization of surface layers. Such a polarization has to remove the inversion center in surface liquid crystalline layers and to allow the SHG to be detectable. Another explanation of the zero-field SHG in terms of the electric quadrupolar interaction was suggested in. ... 

In this section we will consider the orientational deformations of a nematic director which are flexoelectric in nature, i.e., induced due to the interaction of the flexoelectric polarization (see (4.2)) with the external electric field. Our consideration will be limited to spatially uniform fields E the case when E depends on coordinates is discussed in Chapter 5. We also discuss semiphenomenological approaches for the determination of nematic flexoelectric moduli ei and 63. Different types of electrooptical phenomena, where flexoelectric distortion plays a dominant role will be considered. Some of them are promising for potential applications. 

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