Leslie-Ericksen coefficient

Note 2 The friction coefficients can be expressed in terms of the Leslie-Ericksen coefficients as follows ... 

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

In the case under investigations, which includes nematic (anisotropic) phase environments, we shall assume the usual approximation of considering isotropic local friction, and the macroscopic local viscosity is taken equal to half of the fourth Leslie-Ericksen coefficient 1/4 [92-95]. The diffusion tensor of the system is obtained, neglecting translational contributions, as a 4 x 4 matrix, that is. 

In examining three different variants of the position of 71 relative to the directions of the velocity v and velocity gradient y, it is possible to distinguish and experimentally measure viscosity coefficients T i, Ti2. and TI3, which are correlated with the Leslie-Ericksen coefficients [50, p. 200] by the relations... 

Note 3 The Miesowicz coefficients are related to the Leslie-Ericksen coejficients by the following equations ... 

Leslie recognized from early experiments that the anisotropy of the materials calls for multiple viscosity coefficients corresponding to different orientation of the LC relative to the flow. Combining this idea with the Ericksen theory leads to the Leslie-Ericksen (LE) theory, which comprises two elements one describing the evolution of n(r) in a flow field, and the other prescribing an extra stress tensor due to the evolving (r) field. 

The hydrodynamic theory for uniaxial nematic liquid crystals was developed around 1968 by Leslie [10, 11] and Ericksen [12, 13] (Leslie-Ericksen theory, LE theory). An introduction into this theory is presented by F. M. Leslie (see Chap. Ill, Sec. 1 of this Volume). In 1970 Parodi [14] showed that there are only five independent coefficients among the six coefficients of the original LE theory. This LEP theory has been tested in numerous experiments and has been proved to be valid between the same limits as the Navier-Stokes theory. An alternative derivation of the stress tensor was given by Vertogen [15]. 

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]... 

S. Kazunori, G. C. Berry, Frank elastic constants and Leslie-Ericksen viscosity coefficients of nematic solutions of rodlike polymer. Mol. Cryst. Liq. Cryst. 1987,153,133-142. 

Frank Elastic and Ericksen Leslie Viscosity Coefficients... 

The experimental verification of the adequacy of this approach is a matter for the future. At present, not even the Leslie-Ericksen viscosity coefficients (aj-Og) and the so-called Miesowich viscosity coefficients [49], whose physical meaning is clear from Fig. 9.3, have been determined experimentally. 

Here 1111 and are the longitudinal and transverse coefficients of the viscosity of a uniaxial anisotropic liquid which are correlated by linear correlations with the Miesowich and Ericksen-Leslie viscosity coefficients. In addition, these coefficients would be equal to Miesowich viscosities t j and 1)3 if there is an ideally oriented solution in the gap. However, it is practically impossible to obtain the ideal orientation in shear flow. 

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

We shall now discuss the application of the Ericksen-Leslie theory to some practical problems in viscometry. Probably the first precise determination of the anisotropic viscosity of a nematic liquid crystal was by Miesowicz. He oriented the sample by applying a strong magnetic field and measured the viscosity coefficients in the following three geometries using an oscillating plate viscometer ... 

The force g normal to the layers will be associated with permeation effects. The idea of permeation was put forward originally by Helfrich to explain the very high viscosity coefficients of cholesteric and smectic liquid crystals at low shear rates (see figs. 4.5.1 and 5.3.7). In cholesterics, permeation falls conceptually within the framework of the Ericksen-Leslie theory > (see 4.5.1), but in the case of smectics, it invokes an entirely new mechanism reminiscent of the drift of charge carriers in the hopping model for electrical conduction (fig. 5.3.8). 

The molecular approach which we will see eventually proved to be most successful in treating negative is based on the work of Doi [23]. Doi noted that the well established phenomenological theories for thermotropes (which he termed TLP for Ericksen, Leslie and Parodi [68]) which is successful in describing many dynamic phenomena in MLC nematics, is limited for polymeric liquid crystals in that it does not predict nonlinear viscoelasticity. Doi s approach determines the phenomenological coefficients from molecular parameters, so that the effects of, for example, molecular weight and concentration can be treated. He considers a single molecule (the test rod ) and notes that as concentration increases, constraints on its motion are imposed by collisions with other rods. This constraint can be modeled as a tube... 

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) ... 

The hydrodynamic continuum theory of nematic liquid crystals was developed by Leslie [1,2] and Ericksen [3, 4] in the late 1960s. The basic equations of this theory are presented in Vol. 1, Chap. VII, Sec. 8. Since then, a great number of methods for the determination of viscosity coefficients have been developed. Unfortunately, the reliability of the results has often suffered from systematic errors leading to large differences between results. However, due to a better understanding of flow phenomena in nematic liquid crystals, most of the errors of earlier investigations can be avoided today. 

When limiting our attention to low-molecular-weight nematics, we may expect that, in general, flow has the following effects (1) it alters the distribution of molecular orientations about the nematic axis (director) and (2) it affects the director itself. In other words, the velocity v(r) and the director n(r) are coupled under flow of nematic solutions. Next, we first present the expressions for stress, then discuss some important features of the Ericksen-Leslie theory, and finally show relationships existing between the six Leslie coefficients and three molecular parameters appearing in the Doi theory. The presentation of the entire Ericksen-Leslie theory (Ericksen 1960 Leslie 1966, 1968, 1979) is beyond the scope of this chapter. 

Relationships between the Six Leslie Coefficients and Three Molecular Parameters in the Doi Theory By limiting their analysis to the first-order perturbation from the equilibrium state, Kuzuu and Doi (1984) derived the Ericksen-Leslie equation from the Doi theory under weak velocity gradient and obtained the following relationships between the six Leslie coefficients and the three molecular parameters (concentration, molecular weight, and the order parameter) appearing in the Doi theory ... 

---