Viscosity coefficients rotational

Note 1 The rotational viscosity coefficients are of the order of lO -lO" Pa s for low-molar-mass liquid crystals for polymeric liquid-crystals their values depend strongly on the molar mass of the polymer. 

The physical meaning of the above equation is that viscosity torque, which is the product of the rotational viscosity coefficient y and the angular speed dG/dt, is balanced by -SfISG, which is the sum of the elastic and electric torques. Using Equations (5.53), (5.54) and (5.55), it can be obtained that... 

Although the dynamics of Freedericksz transition in splay geometry, bend geometry, and twisted geometry is more complicated, the response time is still of the same order and has the same cell thickness dependence. The rotational viscosity coefficient is of the order O.IN - s/m. When the elastic constant is 10 "N and the cell thickness is 10pm, the response time is of the order 100 ms. Faster response times can be achieved by using thinner cell gaps. 

Recently Tao et al. extended the MS theory by adding to Eq. (3) the isotropic, density-dependent component of the molecular interactions (/o(r) in the form of the Lennard-Jones potential (/o(r) = 4e [(o-/r) -(o-/r) ]. As a result they obtained a better agreement of the calculated and experimental quantities characterizing the nematic-isotropic transition, for example, volume change at and the values of dT ldp. Chrzanowska and Sokalski considered the case when the parameter Lennard-Jones potential is dependent on the orientation of molecules that allows one to predict properly for MBBA such properties as order parameters, elastic constants, and rotational viscosity coefficients. 

Measuring the torque on a sample of a nematic liquid crystal in a magnetic field rotating with an angular velocity smaller than the critical one represents a relatively simple method for the determination of the rotational viscosity coefficient. Below the critical angular velocity Eq. (24) is valid with 0 = F. Neither the phase lag F-0 nor the anisotropy of the magnetic susceptibility have to be known. This method will be thoroughly discussed in Chap. Ill, Sect. 2.6 of Vol. 2A of this Handbook. 

If the director is free to rotate there will be either a rotation to a stable orientation (flow alignment, see Fig. 6) or a continuous rotation (tumbling) under the influence of the shear gradient. Which case is observed depends on the signs of 2 and a. Because of thermodynamical arguments (see Sect. 8.1.9) the rotational viscosity coefficient must be positive. 

For a common liquid crystal with negative 02 and positive a positive angular velocity 0 of the director will be coupled with positive shear gradients at the solid surfaces as the angular velocity vanishes there. This leads to a positive rotation of the bulk and an increase of the angular velocity of the director with respect to a fixed system of coordinates. This effect is usually described by introduction of an effective rotational viscosity coefficient y which is smaller than y. 

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... 

It is often interesting to compare the rotational viscosity coefficient /j in the nematic phase and... 

The nine coefficients can, therefore, be determined from the seven viscosity coefficients, the rotational viscosity coefficient 2A5 and the relaxation time for the flow alignment for y/=0. Instead of the two last determinations it is also possible to use two flow alignment angles at different V values. 

Determination from Shear and Rotational Viscosity Coefficients.165... 

This section covers experimental methods for the determination of shear and rotational viscosity coefficients of monomeric nematic liquid crystals and experimental results on this topic. Polymeric nematic liquid crystals are dealt with in Chap. V in Vol. 3 of this Handbook. 

Figure 9. Shear viscosity coefficients r/i, t 2 and t)3, rotational viscosity coefficient y, and isotropic shear viscosity coefficient as a function of temperature for the liquid crystal Nematic Phase V. T, Clearing point temperature.

The relaxation time constant T is proportional to the rotational viscosity coefficient. Furthermore, it depends on the layer thickness a and the elastic coefficient 22 ot the anisotropy of the magnetic susceptibility and the critical field strength Hf. for this geometry. Two of these quantities have to be determined in a separate experiment. 

There are good reasons for the large interest in the rotational viscosity coefficient y. First, the switching time of displays on the basis of nematic liquid crystals is mainly determined by the rotational viscosity of the liquid crystal used (see Eqs. 46 a and 46 b). Secondly, there is no analogue to the rotational viscosity in isotropic liquids. 

Flgure 17. Rotational viscosity coefficients y, of the homologous di-n-alkyloxyazoxybenzenes as a function of temperature. The numbers on the curves denote the length of the alkyl chain. 

Figure 18. Rotational viscosity coefficient divided by the order parameter squared as a function of temperature for the homologous series of 4-alkyloxybenzyl-idene-4 -n-butyl-anilines (mO 4).

Figure 19. Rotational viscosity coefficient y at 25 °C as a function of the free volume coefficient Vfg = 1 - fcp, where is the molecular packing coefficient (1) al-kyloxycyanobiphenyls (2) alkylbicyclooctylcyano-benzenes (3) alkylpyridylcyanobenzenes (4) alkylcyanobiphenyls (5) alkylcyclohexylcyanobenzenes. The crosses show the rotational viscosity for some members of the homologous series and the lengths of the vertical lines give the accuracy of measurement.

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 study of light scattered by a nematic sample allows the determination of the viscosities 7]j(q) and Tj Cg) (see Eqs. 17 and 18), which are mainly determined by the rotational viscosity coefficient yj, but also contain other coefficients because of the backflow. In principle, all coefficients can be determined with different accuracies by a suitable choice of the scattering geometry. The influence of the small coefficient Oi is normally neglected. For 4-n-pentyl-4 -cya-nobiphenyl, Chen et al. [93] found the following values at 25 °C 03=-0.086, a3=-0.004, a4=0.089, c%=0.059 and o = -0.031 Pa s. 

The rotational viscosity coefficient yi is the most frequently determined viscosity coefficient of liquid crystals. By a straightforward adaptation of the shear flow technique, Tsvetkov originated the idea of measuring a specific torque, TA, exerted on the sample by n rotating with a constant angular velocity (0[8] ... 

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. 

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