The flexoelectric effect was first discussed in 1969  as a liquid crystal analogue to the piezoelectric effect6.1. Flexoelectricity is the generation of a spontaneous polarization in a liquid crystal due to a deformation of the director, or conversely, the deformation of the director due to an applied electric field.
Classically the flexoelectric effect arises from molecules with a shape asymmetry. The first cases to be considered were wedge and banana shaped molecules. Wedge shaped molecules with longitudinal dipoles show spontaneous polarization when splayed. Likewise banana shaped molecules with transverse dipoles exhibit spontaneous polarization under bend deformation. These cases are shown in Fig. 6.1.
Prost and Marceau extended this analysis to more symmetric molecules with permanent quadrupole moments . For quadrupolar molecules a polarization can occur from the gradient of the quadrupolar density. A simple physical picture of this can be seen in Fig. 6.2. In an undistorted state there is no bulk polarization. If the system is splayed, positive charges can enter a layer from above, while they are expelled below. This gives rise to a dipole, hence a spontaneous polarization.
Flexoelectricity is also possible in symmetric polar liquid crystals such as 5CB and PCH5. In this case flexoelectricity arises from the effect that splay and bend deformations have on the local association between molecules. Polar liquid crystals tend to form dimers with anti-parallel alignment between the molecular dipoles [242,93,102]. When these are subjected to splay or bend deformations the alignment is no longer completely anti-parallel leading to a net polarization. This is shown in Fig. 6.3. This dimerization would also lead to the quadrupolar effect, at distance the anti-parallel dipoles would have a similar effect to a quadrupole.
In the above cases the polarization couples to a splay and/or bend
deformation. It can be seen from symmetry arguments that the twist deformation
cannot give rise to a polarization . Thus a
phenomenological formula for the flexoelectric polarization can be written as
The flexoelectric effect has a large influence on many phenomena in liquid crystals . Technologically it plays a key role in some novel device applications. Flexoelectric surface switching is important in newly developed bistable displays [245,246]. Flexoelectric coupling in chiral and twisted nematic crystals  leads to a linear rotation of the optic axis and also leads to device applications . Flexoelectric coupling in smectic liquid crystals has been shown to stabilize helical structures .The flexoelectric effect is also present in lipid membranes . The direct link between molecular structure and the flexoelectric effect also makes it of fundamental interest.
There have been many experimental studies of the flexoelectric effect [250,251] and some experimental techniques for determining the flexoelectric coefficients are outlined in §6.2. However, and are difficult to determine from experiment for a number of reasons. Firstly the response of liquid crystals to applied fields is dominated by the dielectric response for all but the smallest fields. Secondly the flexoelectric coefficients are not measured directly; generally and are measured. Alongside experimental studies, several theoretical studies have also been performed [252,253,254,255,256,257,258,259]. A number of different approaches were used. An Onsager-like theory, a mean-field theory (including attractive and repulsive interactions) , and density functional theories [256,257] have all been developed. By necessity numerical results from these theories have only been determined for simple models of liquid crystals. Recently more sophisticated theoretical studies have attempted to calculate the flexoelectric coefficients using more realistic models of liquid crystals  or to take the effect of intermolecular interactions into account ,
Despite the difficulties in experimental or theoretical approaches only two previous simulation studies of the flexoelectric effect have been performed [76,77,78]. These both used simple models and are outlined in §6.3.