Torques viscous

The fonn of y to be used in Nee and Zwanzigfs equation is the true y in the material, and itself depends on e through the viscous torque experienced by a dipole rotating close to a lossy material. The torque N is related to the angular velocity Q either by... 

A spherical-top molecule need not be geometrically spherical, so there may be three distinct principal axes for the viscous torque on the rotating molecule. Steele thus obtmns three equations of motion... 

From the Ericksen-Leslie theory, we know that the viscous torque is... 

The physical interpretation of this distortion is as follows. (We shall assume throughout this discussion that the material is an ordinary nematic like MBBA with < 0.) Let us suppose that there is a fluctuation 0 > 0 in the xy plane that rotates the director from its initial orientation (1,0,0). The flow now exerts a viscous torque, which is given by SO (fig. 

Fig. 3.11.1. The mechanism for homogeneous instability the flow is along y and the velocity gradient along z. (a) An angular fluctuation 0 = n > 0 results in a viscous torque T, > 0 such that a small twist (j> = n >Q is produced, (b) a deformation > 0 results in a torque < 0 such that tne initial deformation 0 is...

For a given 4> there is the torque S< > due to the primary velocity (as in HI) as well as a torque due to the secondary velocity gradient and together the total viscous torque F, brings about a distortion 0. In turn, a 0 distortion gives rise to a torque that increases . (We have already seen in the case of HI that T = —p S0, but this value is now modified slightly by a torque due to v. )... 

Figure 33 shows that the RPA can be an advanced, roiorless curcnietcr. Both the S elastic torque and S " viscous torque are shown during cure. The S " cure curve is more sensitive to variations in oil and carbon black than the S cure curve [123]. 

In this chapter, we will be describing different electro-optic techniques that have been developed specifically for measurement of the flexocoeffi-cients. In these techniques, an electric field is applied to a nematic liquid crystal in a cell with well-defined boundary treatments. The net torque on the medium, which involves dielectric, elastic and flexoelectric components, is set to zero both in the bulk and at the two surfaces in order to And the equilibrium director configuration if a DC field is applied. In AC techniques, the above torques are balanced with the viscous torque. The optical properties of the medium are in turn calculated for the distorted profiles and compared with experimental measurements. The flexoelectric, and in some... 

Electroconvection in nematics is certainly a prominent paradigm for nonequilibrium pattern-forming instabilities in anisotropic systems. As mentioned in the introduction, the viscous torques induced by a flow field are decisive. The flow field is caused by an induced charge density p i when the director varies in space. The electric properties of nematics with their quite low electric conductivity 10 (fl m) ] are well described within the electric quasi-static approximation, i.e. by charge conservation and Pois-... 

For the parameter combination o < 0 and <7a < 0, which can be found in some nematic compounds, electroconvection is definitely excluded within the standard model. Nevertheless, EC has surprisingly been observed in this case (for recent examples see, e.g. Kochowska et alP and Toth-Katona et al ). The theoretical analysis has proved that flexoelectricity is crucial for understanding this non-standard EC because in Eq. (4.7) the contribution V Pfi to Pel is dominant. It is interesting that the flexotorque on the director is determined by the difference (ei — 63) of the flexocoefScients while the sum (ei - - 63) governs the flexocharge and thus its contribution to the viscous torque. Further details will be sketched in Section 4.3.2. 

As the shear rate increases, large viscous stresses cause refinement of the texture until eventually individual molecules experience significant viscous torques, and the molecular order parameter can be distorted from its equilibrium value. This occurs when the Deborah number approaches or exceeds unity. 

The viscous torque is given by Eq. (9.22). Due to the absence of flow it contains only one term, namely. 

In each particular situation, these torques may be balanced by the elastic, surface or viscous torques. The magnetic and electric field torques may be obtained differently. Using minimisation of the free energy with respect to the director one obtains the molecular field introduced earlier, see Eq. (8.27) and then finds the torques as vector products with the director. Let us show it. The magnetic free energy density is given by... 

It is simpler to examine the dynamics of the Frederiks effect for the experimental geometry of Fig. 11.15c, since a pure twist distortion is not accompanied by the backflow effect (see the next Section). For a twist distortion we operate with the azimuthal director angle tp(z) (sintp riy) and the equation for rotation of the director that expresses the balance of elastic, magnetic field and viscous torques is given by... 

It is the same general equation (9.22) for director motion adapted to our simple situation we have the familiar form (9.32) for the elastic and viscous torques and, in addition, the coupling term with the same coefficient A describing the torque exerted on the director by shear dvjdz. 

As to dynamics of the response of a SSFLC cell to the alternating field, it is controlled by Eq. (13.38) with the viscous j dipldt torque added. When the helical stmcture of the SmC is unwound in a thin cell (a typical case) one can neglect the elastic torque d

anchoring energies Wf and W are reasonably weak, the electric field torque would solely be balanced by the viscous torque ... 

To describe the dynamics of d> at constant 9, the viscous torque with viscosity coefficient y,p should be added to the balance equation... 

Santamato et ai assumed that with homeotropic boundary conditions the normal component of the optical torque cannot be balanced by an elastic torque, hence the molecules are set into rotation. is balanced by a viscous torque arising from the precession of the director. They established a simple relation between the angular velocity of the director rotation and the total change of the Stokes parameter, S3. This relation was experimentally verified. 

We now consider the time dependence of the molecular reorientation. For the total free energy density defined by Eq. (2.1) we include a dissipative term 7/0 /3r)/2 which will contribute a viscous torque opposing any rapid... 

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