Freedericksz transitions

The standard method for measuring elastic constants in nematics is the analysis of onedimensional director deformations in thin planar cells with defined anchoring conditions. This approach allows separate study of splay, twist or bend deformations in a pure form at the onset of the transition. 

It has been known since Frdedericksz s work in the 1930s [7, 12] that the threshold field for director deformations in planar cells can be used to determine the elastic coefficients of nematic liquid crystals. Saupe [13] was the first to describe analytically the static director field in planar cells under the action of an external magnetic field in terms of Frank s elastic theory, and not only did he derive expressions for the threshold mag- 

For the bend transition in homeotropic cells this holds analogously, but and 33 change their roles. 

In these geometries, the sample cell is usually observed in the transmitted light with crossed polarizers inserted in 45° orientation to the director tilt plane. While the ordinary polarized light in the birefringent medium always experiences the ordinary index of refraction and is unaffected by the deformation, the optical path of the extraordinary wave is sensitive to the director tilt. Both waves are brought to interference at the analyser and the optical interference can be related in a direct way to the director deflection. Saupe first used this magneto-optical method with splay geometry to determine the splay and bend constants of p-azoxyanisol (PAA) [13]. 

A prerequisite for the magneto-optical experiments is a knowledge of the diamagnetic anisotropy Bx and the cell thickness, in order to determine the elastic constants from the threshold fields, and the refractive indices, in order to fit the complete transmission characteristics. 

Liquid crystals reorient in externally applied electric fields because of their dielectric anisotropies. The electric energy (a part of the free energy) of a liquid crystal depends on the orientation of the liquid crystal director in the applied electric field. Under a given electric field, the liquid crystal will be in the equilibrium state, where the total free energy is minimized. 

Substitution into Eq. [7] and application of the Euler-Lagrange equation results in 

However, when d/(2 ) tt/2, a second solution with 51 0 is obtained that gives lower free energy than the = 0 solution. The critical magnetic field Hp for the transition is found by equating d/(2 M) 

The right-hand side can be numerically integrated on computer, and the results x) are plotted in Fig. 9 for three different values of H. 

9—TW angle of (he directors plotted as a function of posHlon In a Freedericksz cell for three different magnetic Held strengths. 

The Freedericksz transition can be induced by an electric field as well as by a magnetic field. The threshold electric field in that case is given by 

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If we compare with figure C2.2.I I, we can see that this defonnation involves bend and splay of the director field. This field-induced transition in director orientation is called a Freedericksz transition [9, 106, 1071. We can also define Freedericksz transitions when the director and field are both parallel to the surface, but mutually orthogonal or when the director is nonnal to the surface and the field is parallel to it. It turns out there is a threshold voltage for attaining orientation in the middle of the liquid crystal cell, i.e. a deviation of the angle of the director [9, 107]. For all tliree possible geometries, the threshold voltage takes the fonn [9, 107]... 

Figure C2.2.12. A Freedericksz transition involving splay and bend. This is sometimes called a splay defonnation, but only becomes purely splay in the limit of infinitesimal displacements of the director from its initial position [106]. The other two Freedericksz geometries ( bend and twist ) are described in the text.

Note 1 The Freedericksz transition occurs when the strength of the applied field exceeds a certain threshold value (see Definition 5.12). 

BaTiC>3 particles are another very attractive and intensively studied type of nanoparticles in nematic liquid crystals. Cook et al. reported on an asymmetric Freedericksz transition, where doping nematic TL205 with single domain ferroelectric BaTiC>3 nanoparticles (9 nm in diameter) reduced or increased the threshold voltage by 0.8 V depending on the polarity of the applied voltage [149]. 

Blach and co-workers also observed a lower Freedericksz transition threshold voltage (V ii,) for 5CB doped with rather large BaTi03 particles (150 nm in diameter) [316], which is surprising considering an earlier report by West and Reznikov et al., who found no such reduction of Vth using smaller, chemically similar nanoparticles [317]. 

OPTICAL-INDUCED FREEDERICKSZ TRANSITION OF NEMATIC LIQUID CRYSTAL DOPED WITH PORPHYRINATOZINC( II)... 

The Optical-Induced Freedericksz Transition of Nematic Liquid Crystal (5CB) doped with l%(w/w) of 5, 10, 15, 20-tetraphenylporphyrinatozinc( II) (ZnTPP) were studied. Excited by Ti Sappire laser with the 82MHz repetition rate and lOOfs pulse duration, the optical Freedericksz threshold of a 23.6pm-thickness planar alignment sample occurred at an intensity level of0.35mW/mm in contrast to the normally observed 83mW/mm value for pure 5CB. The coordination-bonding interaction between 5CB and ZnTPP were discussed by UV-vis and fluorescence spectra. We attribute the reduction of the optical Freedericksz threshold to the coordination-bonding interaction. 

Due to their large optical anisotropies, liquid crystals (LCs) have a large optical nonlinearity which is the result of molecular reorientation (Freedericksz transition) in an external field which exceeds the critical field [1], The high external field inhibits the application of LCs, and decreasing the threshold as low as possible is a difficult task [2], LCs doped with a small amount of absorbing dyes that could decrease the needed optical field intensity have been reported [3]. The basic assumption is that the anomalous reorientation of the director results from the interaction between the excited dye molecules and the host. However, this sample would easily degrade under the influence of laser radiation. 

In this paper, we first observed the optical-induced Freedericksz transition of nematic liquid crystal (5CB) doped with 5, 10, 15, 20-tetraphenylporphyrinatozinc(ll) (ZnTPP)... 

Problem 10.1(b) (Worked Example) Calculate the critical magnetic field required to induce the Freedericksz transition described in part (a), where d is the gap between the plates. 

Problem 10.1 (c) Compute the critical magnetic field for a Freedericksz transition in which the director is initially oriented in the z direction, perpendicular to the plates, and strongly... 

Wall defects are also very important in nematic phases, especially in electric or magnetic fields. This will be considered further in section C2.2.4.1. which discusses Freedericksz transitions in a nematic in an electric or magnetic field. 

Here nd are elastic constants. The first, is associated with a splay deformation, K2 is associated with a twist deformation and with bend (figure C2.2.11). These three elastic constants are termed the Frank elastic constants of a nematic phase. Since they control the variation of the director orientation, they influence the scattering of light by a nematic and so can be determined from light-scattering experiments. Other techniques exploit electric or magnetic field-induced transitions in well-defined geometries (Freedericksz transitions, see section (C2.2.4.1I [20, M]. 

In the second group we find pattern forming phenomena based on new instability mechanisms arising from the specific features of liquid crystals, which have no counterpart in isotropic fluids or at least are difficult to assess. Some examples are shear (linear, elliptic, oscillatory, etc.) induced instabilities, transient patterns in electrically or magnetically driven Freedericksz transitions, structures formed in inhomogeneous and/or rotating electric or magnetic fields, electroconvection (EC), etc. [5-7]. 

Figure 2. Threshold voltage Uth/Uo and the critical wavenumber qc versus the dimensionless dielectric anisotropy eajev calculated from Eqs. (7) and (8). a b Planar alignment with a a > 0, c d homeotropic ahgnment with a a < 0. Dashed lines correspond to the Freedericksz transition, solid lines to the direct EC transition.

Figure 7. Threshold voltages Uth/Co and the critical wavenumber qc versus the relative dielectric anisotropy eajei. calculated from Eq. 8. Homeotropic alignment with <7a > 0. The upper (a b ) and lower (c d ) plots differ only in the axis scales. Dashed lines correspond to the Freedericksz transition, solid lines correspond to the direct transition to an ("a-induced") EC patterned state, dotted lines represent a secondary transition to EC.

Case H homeotropic alignment, Ca < 0 era < 0. Above the Freedericksz transition where no standard EC is predicted, convection (ns-EC) builds up with properties similar to those listed for case G. The patterns are disordered (see Fig. 14) as expected for an initial homeotropic alignment. 

In this paper we have reviewed the structures appearing at onset of electro-convection in nematic liquid crystals. The influence of the relevant material parameters (ca and ao) and the role of the initial director alignment were explored. Our calculations using a linear stability analysis of the standard model of electroconvection (performed for zero frequency) revealed that four different scenarios characterized by different ranges of the wavenumber q can be identified (1) the Qf= 0 mode (a homogeneous deformation known as the Freedericksz transition) predicted and observed in cases C, D, E and H, which is... 

The subcritical nature of the Freedericksz transition can be explained as follows. When the director settles to the precession state, light becomes el-liptically polarized inside the nematic. On the other hand, it is known that the Freedericksz transition for elliptically polarized light depends on ellipticity and... 

The Freedericksz transition in the nematic and smectic C phases of 3- -heptyl-6-(4- -hexyloxyphenyl)-l,2,4,5-tetrazine has been studied. In both phases, the threshold voltage and the switching times were measured. In the smectic C phase, two thresholds have been observed which can be explained by an asymmetric chevron structure <1996MI131>. 

The Freedericksz transition discussed in 3.4.1 may be called a homogeneous transition since the distortion occurring above the threshold is uniform in the plane of the sample. In low-molecular-weight nematics, which as a rule have relatively small elastic anisotropy k i kjj 2 22), it is the homogeneous transition that is generally observed. Some polymer nematics, however, are known to exhibit high elastic anisotropy - an example is a racemic mixture of poly-y-benzyl-glutamate (PEG) which has k Jk =11.4 and k /k = 13.0 - and in such cases more complex types of field-induced deformations are possible. ... 

For a concise review of Freedericksz transition in polymer nematics, see U. 

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