Viscosity tensors

Furtlier details can be found elsewhere [20, 78, 82 and 84]. An approach to tire dynamics of nematics based on analysis of microscopic correlation fimctions has also been presented [85]. Various combinations of elements of tire viscosity tensor of a nematic define tire so-called Leslie coefficients [20, 84]. 

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. 

Out of the five viscosities, only two (V2 and V3) show a significant influence on the critical values. In Fig. 8 we present the dependence of 9C and qc on an assumed isotropic viscosity (upper row) and on these two viscosity coefficients (middle and lower row). Since the flow alignment parameter X has a remarkable influence on these curves we have chosen four different values of X in this figure, namely X = 0.7, X = 1.1, X = 2, and X = 3.5. The curves for X < 1 and X > 3 for an isotropic viscosity tensor are very similar to the corresponding curves where only V2 is varied. In this parameter range the coefficient V2 dominates the behavior. Note that the influence of V3 on the critical values is already much smaller than that of V2. We left out the equivalent graphs for the other viscosity coefficients, because they have almost no effect on the critical values. For further comments on the influence of an anisotropic viscosity tensor see also Sect. 3.4. 

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. 

Consequently only five independent coefficients actually exist in the nematic liquid crystals. The viscosity tensor is no longer symmetrical and hence a viscous moment appears... 

Continuum theory has also been applied to analyse the dynamics of flow of nematics. The equations provide the time-dependent velocity, director and pressure fields. These can be determined from equations for the fluid acceleration, the rate of change of director orientation in terms of the velocity gradients and the molecular field, and the incompressibility condition. Further details can be found in de Gennes and Frost (1993). Various combinations of elements of the viscosity tensor of a nematic define the so-called Leslie coefficients. 

The vectors f/ and f are the geometric linear and nonlinear inner forces, respectively. K denotes the stiffness tensor and C denotes the viscosity tensor ... 

Liquid crystals as anisotropic fluids exhibit a wide range of complex physical phenomena that can only be understood if the appropriate macroscopic tensor properties are fully characterized. This involves a determination of the number of independent components of the property tensor, and their measurement. Thus a knowledge of refractive indices, electric permittivity, electrical conductivity, magnetic susceptibilities, elastic and viscosity tensors are necessary to describe the switching of liquid crystal films by electric and magnetic fields. Development of new and improved materials relies on the design of liquid crystals having particular macroscopic tensor properties, and the optimum performance of liquid crystal devices is often only possible for materials with carefully specified optical and electrical properties. 

Figure 76. The principal axes 1, 2, and 3 of the viscosity tensor. In the right part of the figure is illustrated how the distribution function around the director by necessity becomes biaxial, as soon as we have a nonzero tilt 6.

If, on the other hand, we had neglected the small 73 in Eq. (342), the viscosity tensor would have taken the form... 

Using dielectric relaxation spectroscopy, it is possible to determine the values of the rotational viscosity tensor corresponding to the three Euler angles in the chiral smectic C and A phases. These viscosity coefficients (Ye, Yq>, 7i) active in the tilt fluctuations (the soft mode), the phase fluctuations (the Goldstone mode), and the molecular re-... 

Figure 81. The three components of the viscosity tensor measured by dielectric relaxation spectroscopy. The substance is LCl (from Buivydas [155]).

The viscosity tensors may have 81 + 27 elements. However, fortunately only those which are allowed by symmetry will be nonzero. The nonzero elements are determined so that the symmetry of the viscosity tensors must be compatible with the symmetry of the material (for example, a calamitic nematic and smectic A material with uniaxial symmetry will be invariant under n <=> -n ). 

For noncompressible isotropic viscous liquids, the viscosity tensor is... 

For anisotropic liquids, the viscosity tensor can be expressed in terms of second-rank unit tensors Si and the internal parameters (in the general case, ten-... 

The orientation of particles of a liquid is characterized by one unit vector 71 The rheological constants p, t, tij, 112, and X are usually experimentally determined. The differential equation characterizes the change in the orientation of the particles of the liquid caused by flow. The viscosity tensor of a simple Ericksen liquid is... 

The viscosity tensor is more complex for anisotropic liquids with a lower degree of symmetry. 

The models of liquids with anisotropic viscosity are basically used for establishing the features of the rheological behavior of low-molecular-weight liquid crystals. Their anisotropic viscoelastic behavior is the most significant distinctive feature of polymer LC. LC polymers should be considered nonlinear anisotropic viscoelastic liquids. Their anisotropic properties are inadequately characterized by only one viscosity tensor. It is necessary to introduce another relaxation time tensor which will describe the anisotropy of the relaxation properties of LC polymers. The work in this direction has only just begun, and only the basic approaches to the study of the anisotropic viscoelasticity of LC polymers will be reported here. 

The relaxation equation for a Maxwellian incompressible isotropic liquid is obtained from the law of anisotropic relaxation in the following partial form of the relaxation time Xj.ju and viscosity tensors ... 

Due to the symmetry property of crP and Ay, the four-rank viscosity tensor must be Viju = Vjiu = Vijik, and the Onsager reciprocal relations require that Vyu = Vkuy For Harvard notation (7.30) we have... 

Momentum conservation requires that an equal and opposite force be applied to the fluid. Both discrete and continuous degrees of freedom are subject to Langevin noise in order to balance the frictional and viscous losses, and thereby keep the temperature constant. The algorithm can be applied to any Navier-Stokes solver, not just to LB models. For this reason, we will discuss the coupling within a (continuum) Navier-Stokes framework, with a general equation of state p p). We use the abbreviations for the viscosity tensor (46), and... 

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