Periodic Distortions In Nematics Caused By Flexoelectricity And Saddle-Splay Elasticity

In liquid crystal materials, the coupling between their elastic field and an external action, such as electric and magnetic fields or the confinement created by free surfaces or cell walls, can give rise to periodic domains. Here, some simple calculations are proposed for nematics in planar cells, where undulations are caused by flexoelectricity and saddle-splay elasticity.

The first term does not depend on the director field. For this reason, it is not considered in the distortional contribution to the bulk free energy density. Besides director, there is another vector, obtained from its derivatives, which can be used. Let us define it as vector A  is the sum of two vectors: one has the magnitude given by the divergence of the director, the other is the cross product of the director with its rotor. We can find 3

, A A  
in the contribution to the bulk free energy density of flexoelectricity: Flexoelectricity is a property of liquid crystals, similar to the piezoelectric effect. In certain anisotropic materials, which contain molecular asymmetry or quadrupolar ordering with permanent molecular dipoles, an applied electric field may induce a distortion of the director orientation. Conversely, any distortion induces a macroscopic polarization within the material. The polarization vector P  in the flexoelectric term can be described as: In (6), we have used vectors , which can be defined as distortional Lifshitz vectors [11,12]; in [11] we have discussed the role of these vectors and helicity in the nematic free energy density.
The coupling of polarization P  with an external electric field results in the appearance of a periodic distortion from an initial planar orientation of the nematic cell [7]. Meyer showed that the infinite liquid crystal must be disturbed, the perturbation being periodic along the director orientation and the period is inversely proportional to electric field strength [13]. Flexoelectricity in liquid crystals is analogous with piezoelectricity in solids. In the piezoelectric materials, an applied uniform strain can induce an electric polarization and vice versa. Some crystallographic considerations restrict this property to non-centrosymmetric systems.

Periodic distortions in nematics
Let us discuss the results from [7,13], that is the periodic distortion in the infinite medium caused by flexoelectricity. The free energy density is given by: in the uniform elastic approximation with K elastic Let us consider the director n  in a uniform configuration, as a vector parallel to x-axis and the electric field E  parallel to z-axis as   , with a period inversely proportional to the electric field strength. The free energy density of the distorted configuration, including the flexoelectric term, is: Then, a periodic distortion in a non-confined nematic is possible because it has a free energy density lower than that possessed by the uniform configuration. Let us note that there is no threshold for the electric field. Even a small field gives rise to the distorted configuration. The existence of a threshold is a consequence of the medium confinement. Let us imagine a nematic material confined in a cell composed by two plane walls, both parallel to [x,y] plane, at a distance d. The anchoring conditions must be included in the energy balance. We can assume a surface energy density of the Rapini-Papoular , for a surface treatment favouring a molecular alignment parallel to x-axis. If the director field n  is uniform in the planar alignment, Let us assume, as in Ref. [7], the behaviour of the tilt angle in the form . We can integrate the free energy density on a volume V given by where d is the cell thickness, L a fixed length in y-direction and  the director distortion wavelength along the x-direction, we obtain: The last term in (10)   If the electric field has a value * E E  , the stable configuration of the director field is that with lower energy: in this case, it means when the director field is uniform. When * E E  , the stable configuration is the distorted one. Comparing the two values of the total energy, that is: we can approximately find the threshold electric field as: , to have a real electric field: The threshold field has a value:

Flexoelectricity and hybrid cell
Let us consider the role of flexoelectricity in a hybrid nematic cell. This is a cell where the nematic is confined between two parallel walls with different anchoring conditions. One surface is treated to favour planar alignment; the other is favouring homeotropic alignment. The cell is then known as HAN, Hybrid Aligned Nematic, cell. The hybrid cell we discuss has the z-axis perpendicular to cell walls ( Figure 3). An electric field can be applied parallel to z-axis: we have is the unit vector of z-axis. k  is the homeotropic direction too. The unit vector i  , parallel to the cell walls, gives the easy planar direction. The bulk free energy density is given, in the elastic isotropic approximation, by: where the last term is due to the dielectric anisotropy   of the nematic.    under this value of the electric field, it is favoured the planar configuration, over the threshold value, it is the homeotropic configuration that has a lower energy. In a hybrid cell, the director changes from a planar configuration at one of the cell wall, to a homeotropic configuration at the other cell wall. The tilt angle is then depending on z, as a function The director field is given by: If the anchoring is strong, the tilt angle is We observe two threshold fields: when the field is lower than E' , the nematic is planar, if the field is comprised between E' and ' E' , the cell is hybrid. Above the second threshold ' E' , the cell is homeotropic. As previously discussed, the electric field can be coupled with a polarization coming from an elastic deformation in the flexoelectric effect. In planar and homeotropic configurations, the director is uniform and therefore the flexoelectric effect is absent. In the hybrid cell, the deformation exists and gives a flexoelectric polarization are changed from the contribution of the flexoelectricity. They could be lowered or raised by the induced polarization (see Figure 6).   [15,16]. Even a giant flexoelectricity has been found with bent-core nematics: a peak of m nC 35 was measured in these materials then more than 3 orders of magnitude larger than in calamitics [17].

Saddle-splay elasticity, PHAN cell and threshold thickness
In nematics, a more general form of the distortion freeenergy density, in the framework of the usual first-order continuum theory, is given as: The last term in (28) is the contribution of the saddle-splay elasticity [4,5]. In fact, this contribution is not usually inserted in the bulk free energy, because it becomes a surface contribution when integration is made on the cell thickness. In addition to the anchoring energy then, there is an elastic contribution too. Saddle-splay contributions are relevant in evaluating the elastic contribution of thin films or membranes [12,18]. Sometimes, periodic distortions of the director in the HAN cells are observed [4,5]. Because of this periodic configuration, the cell is in a PHAN configuration, that is a nematic cell with a period hybrid alignment. Two angles describe the PHAN configuration: θ and φ. The last angle is formed by the projection of the director in the plane of the cell with the x-axis. The frame of reference is  