The Ericksen-Leslie theory

The nematic liquid crystal differs from a normal liquid in that it is composed of rod-like molecules with the long axes of neighbouring molecules aligned approximately parallel to one another. To allow for this anisotropic structure, we introduce a vector n to represent the direction of preferred orientation of the molecules in the neighbourhood of any point. This vector is called the director. Its orientation can change continuously and in a systematic manner from point to point in the medium (except at singularities). Thus external forces and Adds acting on the liquid crystal can result in a translational motion of the fluid as also in an orientational motion of the director. 

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The Ericksen-Leslie theory is valid for the polymeric liquid crystals if the velocity gradient is small. The theory was applied to examine the director tumbling in liquid crystals and the associated effects. It is concluded... 

The hydrodynamics of the liquid crystalline polymer is described by the Ericksen-Leslie theory but liquid crystalline polymers have their polymeric characteristics, such as the viscosity s dependence on the molecular length. 

As elaborated above, the Ericksen-Leslie theory takes into account only the first rank effect of the velocity gradient of liquid crystalline polymers. If the amplitude of the velocity gradient is high enough, these theories are no longer valid and the steady viscosity of liquid crystalline polymers... 

The negative first normal stress difference under a medium shear rate, characterized by liquid crystalline polymers, makes the material avoid the Barus effect—a typical property of conventional polymer melt or concentrated solution, i.e., when a polymer spins out from a hole, or capillary, or slit, their diameter or thickness will be greater than the mold size. The liquid crystalline polymers with the spin expansion effect have an advantage in material processing. This phenomenon is verified by the Ericksen-Leslie theory. On the contrary, the first normal stress difference for the normal polymers is always positive. 

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 ... 

From the Ericksen-Leslie theory, we know that the viscous torque is... 

We shall now show that the essential features of Helfrich s model can be derived on the basis of the Ericksen-Leslie theory. ... 

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 Ericksen-Leslie theory will hold for the polymeric nematics if the velocity gradient is small. Indeed the singular behaviour in the first normal stress difference is predicted by this theory. ... 

In this section we shall show how such constitutive equations can be derived from the molecular theory given in the previous section. For the sake of simplicity, we first consider the case that the system is homogeneous and there is no magnetic field (so that A = 0 in the Ericksen-Leslie theory). 

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) ... 

In the Ericksen-Leslie theory, the viscous tress tensor is given by... 

In the experimentally typical cases R -500 pm and P 1 pm, which gives the experimentally observed high viscosity at low pressures. Later it was shown that the essential features of the Helfrich model can be explained also on the basis of the Ericksen-Leslie theory. °... 

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. 

Director Tumbling A number of researchers (Carlsson 1984 Kuzuu and Doi 1984 Pieranski and Guyon 1974 Semenov 1983) have investigated, with the aid of the Ericksen-Leslie theory, shear flow of nematic liquid crystals and found that instability... 

The Larson-Doi theory predicts qualitatively the essential features of transient shear flow of some model TLCPs (Ugaz 1999). It is, however, not clear to what extent the Larson-Doi theory can describe the dynamics of TLCPs that do not exhibit tumbling. This is because the Larson-Doi theory is based on the Ericksen-Leslie theory, which determines structural responses through the tumbling parameter A,. As will be presented later in this chapter, the experimental data available to date suggest that TLCPs are flow aligning. It is fair to state that the theoretical attempts reported thus far contain, understandably, many crude approximations, and so do not warrant quantitative comparison with experimental results for textured LCPs, particularly TLCPs. Thus the development of a molecular viscoelastic theory for textured LCPs is still in its infancy. 

In 1992 Leslie [175] published an alternative derivation of the Ericksen-Lesfie theory in the isothermal and incompressible case, which led to a simpler presentation of the results originally derived by Ericksen [73] and Leslie [162, 163]. It is this more recent approach that we partly adopt here it is a more concise exposition of the original theory and allows the derivation of the constitutive theory to be discussed in the more traditional continuum theory variables such as the rate of strain tensor and the relative angular velocity without recoiurse to generalised forces. The Ericksen-Leslie theory will be derived in the next Section and a convenient... 

The Ericksen-Leslie theory from Section 4.2.5 will be used with all director gradients and the elastic energy being set to zero, so that we are dealing with an anisotropic fluid. Incorporating the gravitational potential the relevant dynamic equations... 

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