Director tensor

The transmitted light intensity was calculated by the Berreman 4x4 matrix method [22]. The director tensor is determined by the director tilt angle 7 and the director twist angle (f), which are expressed as... 

An aligned monodomain of a nematic liquid crystal is characterized by a single director n. However, in imperfectly aligned or unaligned samples the director varies tlirough space. The appropriate tensor order parameter to describe the director field is then... 

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

Another rotational diffusion model known as the anisotropic viscosity model156,157 is very similar to the above model, and its main feature is to diagonalize the rotational diffusion tensor in the L frame defined by the director. A similar (but not the same) expression as Eq. (71) is J R(r)co)... 

Deformation leading to a change in the director, where the distortion is described by a tensor of third rank... 

Here 0dm is the angle between the director and the largest component, eq, of the electric field gradient tensor, and rj is the asymmetry parameter. The different coordinate systems used are shown in Figure 1. 

As mentioned in Sect. 2.1, we consider a shear induced smectic C like situation (but with a small tilt angle, i.e., a weak biaxiality). We neglect this weak biaxiality in the viscosity tensor and use it in the uniaxial formulation given above (with the director h as the preferred direction). This assumption is justified by the fact that the results presented in this chapter do not change significantly if we use p instead of h in the viscosity tensor. 

In Fig. 8 we have illustrated that a small viscosity coefficient V2 facilitates the onset of undulations. In this section we will have a closer look at the effect of an anisotropic viscosity tensor and ask whether undulations can be caused only due to viscosity effects without any coupling to the director field (i.e., we consider standard smectic A hydrodynamics in this section). 

Consequently, a parallel alignment of smectic layers is linearly stable against undulations even if the perpendicular alignment might be more preferable due to some thermodynamic considerations. As we have shown in Fig. 8, this rigorous result of standard smectic A hydrodynamics is weakened in our extended formulation of smectic A hydrodynamics. When the director can show independent dynamics, an appropriate anisotropy of the viscosity tensor can indeed reduce the threshold values of an undulation instability. 

Anisotropic fluids, of which nematic liquid crystals are the most representative and simplest example, are characterized by an anisotropic dielectric permittivity. The nematic phase has D,yuh symmetry, and in a laboratory frame with the Z axis parallel to the C , symmetry axis (the director) the permittivity tensor has the form ... 

Recently, the surface tensor model has been used together with the dielectric continuum model to calculate the orientational order parameters of solutes in nematic solvents [8,9,27], Figure 2.32 shows the theoretical results for anthracene and anthraquinone in nematic solvents with different dielectric anisotropy. Considering only the surface tensor contribution, positive Szz and Sxx and negative are obtained, with Szz > Sxx > Syy. This corresponds to what could be expected on the basis of the molecular shape the long axis (z) is preferentially aligned with the director, and the normal to the... 

The residual anisotropy A describes the orientation assumed by heavy water molecules. It is expressed as a function of the Saupe order tensor and of the Euler angles describing the orientation of the D20 water molecule with respect to the local director (29) ... 

Such order can be described in terms of the preferential alignment of the director, a unit vector that describes the orientation of molecules in a nematic phase. Because the molecules are still subject to random fluctuations, only an average orientation can be described, usually by an ordering matrix S, which can be expressed in terms of any Cartesian coordinate system fixed in the molecule. S is symmetric and traceless and hence has five independent elements, but a suitable choice of the molecular axes may reduce the number. In principle, it is always possible to diagonalize S, and in such a principal axis coordinate system there are only two nonzero elements (as there would be, for example, in a quadrupole coupling tensor). In the absence of symmetry in the molecule, there is no way of specifying the orientation of the principal axes of S, but considerable simplification is obtained for symmetric molecules. If a molecule has a threefold or higher axis of symmetry, its selection as one of the axes of the Cartesian coordinate system leaves only one independent order parameter, with the now familiar form ... 

We will assume the mesophase director to be parallel to the direction of the static magnetic field. In the last section, III.E, the no-inertia assumption will be rejected and the diffusion operator (2.6) replaced with the complete Hubbard operator. We shall investigate the spectroscopic effects of molecular inertia in the case of an axially symmetric g-tensor. 

Let us define the z direction to be the orientation direction of the director n. Then the order-parameter tensor S is... 

The term director is usually only defined in the limit of vanishing shear rate where the order parameter tensor has uniaxial symmetry. Even at finite shear rates, however, the order parameter tensor S remains well-defined it can be represented pictorially by an ellipsoidal shape, whose major axis is the largest eigenvalue of S. If, for example, the major axis of S lies in the x y plane, then its angle 9 measured clockwise with respect to the x direction is given by... 

If we now generalize the definition of director to be the unit vector parallel to the major axis of the order parameter tensor, we find that at vanishingly low shear rates, where the order parameter tensor is nearly uniaxial, this definition reduces to the usual meaning of the term director. ... 

A system with an additional axis of symmetry, for example X3, is isotropic. The components of the stress tensor in the system of coordinates x[, x, can be obtained from those in the reference frame of the coordinate system xi, X2, X3 by means of the following cosine directors ... 

We have now reviewed most of the theory necessary for the evaluation of transport coefficients of liquid crystals. We are going to start by showing how the thermal conductivity can be calculated. In a uniaxially symmetric system this transport coefficient is a second rank tensor with two independent components. The component An n relates temperature gradients and heat flows in the direction parallel to the director. The component Aj j relates forces and fluxes perpendicular to the director. The generalised Fourier s law reads... 

Note that A with two subscripts denotes the thermal conductivity which is a second rank tensor and A with one subscript denotes the director constraint torque which is a pseudo vector.) Heat conduction is a dissipative process. The entropy production per unit time and unit volume, a, caused by the heat flow... 

We find that all the elements of the symmetric traceless pressure and the antisymmetric pressure are linear functions of sin26 or cos20. One can consequently calculate all the shear viscosities and all the twist viscosities by using the director constraint algorithm to fix the director at various angles relative to the stream lines and calculating the pressure tensor elements as function of... 

It is often preferable to evaluate 6q by EMD methods because the director fluctuates around the preferred orientation in a shear flow simulation, which makes it hard to obtain accurate estimates. If one performs such a simulation one must fix the director at several alignment angles and calculate the antisymmetric pressure tensor, which, according to Eq. (4.10e), is a linear function of cos 26. One can fit a straight line to the data points and the zero gives... 

These results can be cross checked by performing a simulation where the director is constrained to lie in the vorticity plane but leaving it free to select the alignment angle. The angular distribution of the director is shown in Fig. 5 In these simulations only 256 particles were used. Therefore the distribution is fairly wide. As the system size increases the distribution becomes narrower and it is completely sharp in the thermodynamic limit. The maximum of the distribution appears ai 6 - 20° which is in agreement with the zero of the antisymmetric pressure tensor. A similar value of 6q was also found by using the equilibrium fluctuation relations (4.13) and (4.14). One can consequently conclude that the liquid crystal is flow stable. 

We can identify four pairs of thermodynamic forces and fluxes, the symmetric traceless strain rate (Vu) and the symmetric traceless pressure tensor, the director angular velocity relative to the background, (l/2)Vxu-I2 and the torque density X, the streaming angular velocity relative to the background (l/2)Vxu- and the torque density and the trace of the strain rate V-u and difference between the trace of the pressure tensor and the equilibrium... 

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