Leslie rotational viscosity

Table 2. Selected rotational viscosities and Leslie viscosities ui of the liquid crystalline n-alkyl-cyanobiphenyl series (nCB, n 5-8) obtained by FC flow experiments at di rent temperatures 6kT=Tc T relative to the nematic-to-isotropic transition temperature Tq (clearing point). Within the large error estimations of 10% for yj and 2, 100% for (04 + 05), 300% for 03, and 400% for oi, the results are essentially consistent with data reported in the literature. ...

Several methods have been developed to evaluate the Leslie viscosity coefficients described in detail in [18, 28, 31]. These methods include the inelastic scattering of light [60, 93], pulse [94], and rotating [95] magnetic fields, attenuation of the ultrasound shear wave [96], etc. The results obtained by different methods for such important coefficients as rotational viscosity agree fairly well with each other [78], Fig. 2.25. The simplest and most useful methods for measuring 71-values are based on the dynamics... 

This equation describes the director rotation in a magnetic field H with the inertia term I d ipldt being disregarded, 71 = as — a2 is rotational viscosity, and ai are Leslie coefficients. Equation (4.28) in the limit of small ip angles, ip 1, transforms to... 

The six Leslie coefficients can not be measured directly. They can only be determined with the aid of several experimental methods which ususally lead to combinations of these coefficients. Taking into account the Parodi equation, the six coefficients can be obtained from five linear independent viscosity coefficients. Thus, the four viscosity coefficients rji, rj2 and t/j2 and the rotational viscosity coefficient /] give a, = r)i2... 

This equation describes director rotation in the magnetic field with the inertial term being disregarded, /i=a3-a2 is the rotational viscosity, are Leslie s coefficients. In the limit of small 0 angles, 0-< 1, Eq. (30) reduces to a linear form ... 

If the applied magnetic field greatly exceeds the critical strength, the relaxation will be non-exponential. The hyperbola rotation must then be calculated by means of the Leslie-Ericksen equations, and the rotational viscosity Yi is determined by a fit of the calculated hyperbola rotation to the observed one. A disadvantage of this method is that it is difficult to follow the rotation of the interference figure by eye or by means of automatic equipment. 

The shear viscosity coefficients t], 1)2, V3 and 77i2 and the rotational viscosity coefficient 7i form a complete set of independent coefficients from which the Leslie coefficients can be determined with the help of the Parodi equation. The corresponding equations are given in Chap. VII, Sec. 8.1 of Vol. 1. Figure 24 [74] shows the Leslie coefficients for MBBA as a function of temperature. Due to the different dependence on the order parameter (see Chap. VII, Sec. 8.1 of Vol. 1 of this Handbook), the coefficients exhibit different bending above the clearing point. The temperature dependence of differs greatly from that of the other coefficients, as it is not a real viscosity. 

The values of the rotational viscosity coefflcients obtained for polymers X in the nematic phase (10-10 Pa sec) [37, 40], for polymer vn in the reentrant nematic phase (=5 10 Pa-sec) [42], and finally, the values of the Leslie viscosity coefficients (03, for polymer I (=10 Pa-sec) [43] also indicate the participation of the main chains of the macromolecules in orientational motion. It is evident that the polymeric viscosity of LC melts of comb-shaped polymers also determines all of the basic kinetic features of the orientational processes in external fields. 

When the deforming field is rapidly switched on and off, the transient behaviour of n that follows is determined by the viscoelastic properties of the sample, the boundary conditions, and the initial and final states of the director pattern. Such experiments typically provide the most reliable information on the rotational viscosity coefficient. In order to model transient behaviour in a particular geometry a set of the Leslie equations of motion is solved. This solution gives the time evolution of the azimuthal, 9(t,r), and polar, S(t,r), angles describing the orientation of n with respect to some reference frame at any given arbitrary position r in the sample. These functions are parametrised by the Leslie viscosity parameters and the elasticity constants. 

Table 4.1 The principal viscosities in nematics in terms of the Leslie viscosities. The Miesowicz viscosities and the rotational viscosity make up a canonical set of five independent viscosities.

It will be shown on page 304 in Section 6.3.3 that A5 can be identified as a rotational viscosity that is a priori non-negative this viscosity bears some similarities to the rotational viscosity 71 from the theory of nematics (2A5 plays the role of 71 in simple geometrical set-ups). From the physical point of view, it appears that A5 is the most important viscosity. It is related to the azimuthal rotation of the usual director n as it moves around the smectic cone, as depicted in Fig. 6.1. The other two inequalities in (6.232) and (6.233) above have been derived by Gill and Leslie [113, p.l910. ... 

Figure 10.7 (a-c) The Leslie viscosities 0 3 and o 2 determine the direction and rate of rotation of the director (represented by the cylinders) in the orientations shown, For negative values of 3 and 2 (the usual signs for rod-like nematics), the rotation directions are shown by the arrows. The viscosity ua, determines the viscosity of the liquid when the director is in the vorticity direction. (Adapted from Skarp et al., reprinted with permission from Mol. Cryst. Liq. Cryst. 60 215, Copyright 1980, Gordon and Breach Publishers.)... 

An explanation for these various discrepancies was suggested [Yao and Jamieson, 1998], based on the notion that when the nematic director of the solvent is allowed to rotate, one must take account of the coupling between the solvent director and the LCP director. This induces an additional viscous dissipation mechanism which contributes to the Leslie viscosities and the twist viscosity, but not to the Miesowicz viscosities ... 

The coefficients of friction for the director have the dimensions of viscosity and are particular combinations of Leslie coefficients, ji = aj,- (I2, Ji = 3 + 2-It is significant that only two coefficients of viscosity enter the equation for motion of the director. One (72) describes the director coupling to fluid motion. Eor example, if the director tumes rapidly under the influence of the magnetic held, then, due to friction, this rotation drags the liquid and creates flow. It is the backflow effect that will be described in more details in Section 11.2.5. The other coefficient (Yi) describes rather a slow director motion in an immobile liquid. Therefore, the kinetics of all optical effects caused by pure realignment of the director is determined by the same coefficient yj. However a description of flow demands for all the five viscosity coefficients. 

We use here the Ericksen-Leslie continuum theory to describe the effect. The rotational motion of the director (i.e., molecular reorientation) is driven by the pump laser pulse, but it is also coupled with the translation motion (flow) of the fluid through viscosity. Thus, with a finite pump beam, a rigorous theoretical calculation would require the solution of a set of coupled three-dimensional nonlinear partial differential equations for the angle of... 

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