Magnetic Field-Induced Director Deformation

The free energy density of a liquid crystal in the presence of external bulk interaction is  

Assuming that the magnetic field makes an angle 6 with the director and causes a deformation along the z axis, the Euler-Lagrange equation in the bulk (see Chapter 4) reads as  

This last equation can also be considered as the balance of the elastic and magnetic torque densities, where the magnetic torque density is defined as  

For uniaxial materials characterized by a director n, Tu, can be expressed with the diamagnetic anisotropy Xa/ and the angle between the director 8 and the amplitude of magnetic field H as  

We define the magnetic coherence length as With tiiis definition (7.17) looks like  

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The magnetic field-induced director deformations involving bend or splay are illustrated in Figure 7.5b. 

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]. 

It is convenient to change the variable z in Eq. (72) to one involving the director deformation 0, and for a magnetic field-induced Freedericksz transition, Eq. (57) can be... 

The optical-field-induced Freedericksz transition for a twist deformation by a normally incident laser beam in a planar-aligned nematic liquid crystal is studied. The Euler equation for the molecular director and the equations describing the evolution of the beam polarization in the birefringent medium are solved simultaneously in the small-perturbation limit. The stability of the undistorted state is investigated. An alternate series of stable and unstable bifurcations is found. This phenomenon has no analog in the Freedericksz transition induced by dc electric and magnetic external fields. 

The complication associated with electric fields is due to the large anisotropy of the electric permittivity, which means that above threshold the induced electric polarization is no longer parallel to the applied field. In a deformed sample the director orientation is inhomogeneous through the cell, and as a consequence the electric field is also nonuniform. An additional problem can arise with conducting samples, for which there is a contribution to the electric torque from the conductivity anisotropy. Neglecting this, the expressions for threshold electric fields are similar to those obtained for magnetic fields ... 

At present, at least three types of steady-state dielectrically driven pattern are known for nematics. The electric-field-induced periodic bend distortion in the form of parallel stripes has been observed in a homeotropi-cally oriented layer of 5-CB ( a= 13) in the presence of a stabilizing magnetic field [75, 76]. The stripes with a wavevector q were parallel to the electric field E and stationary at low fields. It was shown that a stable periodic pattern of the director minimizes the free energy of the cell when the ela.stic moduli and A 33 are similar to each other. In these experiments the Frederiks transition is of first order, the nonde-formed and deformed areas coexist at a given voltage, and the front between them may propagate along the direction y perpendicular to both fields [77]. 

In order to relate the cell capacitance or optical transmission curves in splay or bend geometry to the elastic ratios, exact anal3rt-ical equations for the director field in the cell are used. The one-dimensional deformation of the director field n=[cos 6(z), 0, sin0(z)] induced by electric or magnetic fields in splay geometry is found by minimizing the free energy... 

Elastic deformation of the director, induced by a magnetic induction or electric field, in a uniformly aligned, thin sample of a nematic confined between two surfaces. 

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