Leslie coefficients

A main problem in evaluating the divergence of the rotational viscosity in the vicinity of a smectic phase is the separation of the non-divergent part. An additional determination of the pressure dependence can be very advantageous. First, the pressure dependence can be studied directly, as the pressure dependence of the non-divergent part is less pronounced than its temperature dependence. Secondly, the above discussed 

The six Leslie coefficients i to Og are the material constants in the stress tensor of the Leslie-Ericksen equations (see Chap. VII, Sec. 8.1 of Vol. 1 of this Handbook). The coefficients must be known for any calculation of flow phenomena and director rotations by means of the Leslie-Ericksen equations such as, for example, for the prediction of the transmission curve during the switching of a liquid crystal display. Because of the Parodi equation [90] 

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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]. 

Figure 4 shows the director profile at various ydv idy values. Figure 5 shows the corresponding effective viscosity values. Only one set of Leslie coefficients are presented, so no general conclusions can be drawn from the data on the rela-... 

Note 2 For nematics formed by low-molar-mass eompounds, the Leslie coefficients are typically in the range 10 to 10 Pa s. 

Leslie coefficients, Leslie-Ericksen coefficients dielectric anisotropy... 

Figure 6.10. Physical meaning of the a terms of Ericksen-Leslie coefficients. (Modified from Dubois-Violette et al., 1978.)...

The detail of the Doi theory can be refered to in the book by Doi and Edwards The Theory of Polymer Dynamics (1986) in which the Leslie coefficients are related to the order parameter and the isotropic viscosity fj as follows... 

In reality, it is difficult to measure the six Leslie coefficients. However, it may be easier to measure the Miesowicz viscous coefficients which are related to the order parameter and rj... 

Meyer (1982) predicted that as the length of the chain approaches infinity, the Leslie coefficients ai and 2 should tend to —oo and a3 tends to oo, while 04,05 and 06 are of finite values. The Miesowicz viscosities r/a and rjb are finite while r/c tends to infinity because the velocity is perpendicular to the director and the shear flow. 

The viscous stress tensor in Eq. (1) is written in terms of the six Leslie coefficients at [6],... 

The hydrodynamic equations of the classical nematic ( 3.1) are applicable to the N, phase as well. There are six viscosity coefficients (or Leslie coefficients) which reduce to five if one assumes Onsager s reciprocal relations. A direct estimate of an effective value of the viscosity of from a director relaxation measurement indicates that its magnitude is much higher than the corresponding value for the usual nematic. 

We have already seen in 3.6.4 that in ordinary nematics of rod-like molecules, the Leslie coefficients and are both negative and pj > l/ig]. Under planar shear flow, the director assumes an equilibrium orientation 00 given by... 

We have seen that the constitutive equation given by eqns (10.75) and (10.78) agrees with the special case of the Ericksen-Leslie theory. Therefore, by comparing the two equations, it is possible to express the Leslie coefficients by molecular parameters. To carry out this programme, however, we have to consider the situation with both magnetic and velocity gradient fields. If we repeat the same calculation as in Section 10.5.3, we have the following equation instead of eqn (10.114) ... 

Experimentally it is not easy to measure all of the Leslie coefficients. What is often measured is the Miesoviez viscosity defined by... 

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. 

It would be very instructive to relate the experimental (Miesowicz) and theoretical (Leslie) coefficients of viscosity. Our task now is to use the viscous tensor (9.20) and find the relationships between the coefficients for each of the three basic orientations of the director, namely = 1, iiy = 1, or = 1. At first, we shall prepare some combinations of parameters useful in all the geometries mentioned ... 

New viscosity coefficients pj and A, are related to Leslie coefficients. In particular, po = 0 4 (viscosity of an isotropic liquid). Viscosimetry of SmA liquid crystals is difficult. For instance, in geometry (a), the upper and lower plates should be parallel with a great accuracy (few nanometers) otherwise defects appear. However, for several compounds the correspondent viscosities have been measured. In geometry (b) there was found a shear rate threshold above the threshold the isotropic behaviour (04) was observed. At lower rates, defects control a flow. 

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