Leslie-Ericksen viscosity coefficients

Frank Elastic and Ericksen Leslie Viscosity Coefficients... 

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. 

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. 

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 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. 

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. 

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. 

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... 

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. 

---