Ericksen-Leslie

The viscous stress of liquid crystals is defined as the Ericksen-Leslie stress tensor and is related to A, n and w or IV ... 

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

Figure 6.10 schematically shows the physical meaning of the six terms of the Ericksen-Leslie stress tensor. The correspondent flow fields and the viscous moments are depicted. It is seen that a4 is the term arisen from the conventional fluid. ot is symmetrical, representing an extension effect caused by a non-rotation flow, which produces a viscous stress while being without moment. arotation flow which produces a moment, as and a6 terms are non-symmetrical, same as a2 and 03, but they are associated with the stress and moment resulting from nonrot at ional flow. 

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 If the amplitude of the velocity gradient is high enough, these theories are no longer valid and the steady viscosity of liquid crystalline polymers...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... 

The application of the Ericksen-Leslie equations to cholesteric flow is less straightforward than in the case of nematics and no detailed solutions have so far been possible even for simple geometries. However, the behaviour in certain limiting situations can be explained qualitatively. 

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 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... 

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) ... 

The static continuum theory of elasticity for nematic liquid crystals has been developed by Oseen, Ericksen, Frank and others [4]. It was Oseen who introduced the concept of the vector field of the director into the physics of liquid crystals and found that a nematic is completely described by four moduli of elasticity Kn, K22, K33, and K24 [4,5] that will be discussed below. Ericksen was the first who understood the importance of asymmetry of the stress tensor for the hydrostatics of nematic liquid crystals [6] and developed the theoretical basis for the general continuum theory of liquid crystals based on conservation equations for mass, linear and angular momentum. Later the dynamic approach was further developed by Leslie (Chapter 9) and nowadays the continuum theory of liquid crystal is called Ericksen-Leslie theory. As to Frank, he presented a very clear description of the hydrostatic part of the problem and made a great contribution to the theory of defects. In this Chapter we shall discuss elastic properties of nematics based on the most popular version of Frank [7]. 

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

A stable flow alignment, at small shear rates, exists for Aeq l only. For Aeq < 1 tumbling and an even more complex time dependent behavior of the orientation occur. The quantity Aeq - 1 can change sign as function of the variable cf. Fig. 4. For Aeq < 1 and in the limit of small shear rates 7, the Jeffrey tumbling period [18] is related to the Ericksen-Leslie tumbling parameter Agq by... 

The dynamics of optical reorientation in nematics has been studied much less extensively than the steady-state effects. The theoretical description of transient phenomena can be given in the framework of the non-equilibrium version of the continuum theory (Ericksen-Leslie hydrodynamic theory). 

Liquid crystals are generally characterized by the strong correlation between molecules, which respond cooperatively to external perturbations. That strong molecular reorientation (or director reorientation) can be easily induced by a static electric or magnetic field is a well-known phenomenon. The same effect induced by optical fields was, however, only studied recently. " Unusually large nonlinear optical effects based on the optical-field-induced molecular reorientation have been observed in nematic liquid-crystal films under the illumination of one or more cw laser beams. In these cases, both the static and dynamical properties of this field-induced molecular motion are found to obey the Ericksen-Leslie continuum theory, which describe the collective molecular reorientation by the rotation of a director (average molecular orientation). 

We use here the Ericksen-Leslie continuum theory to describe the effect. The rotational motion of the director (i.e., molecular reorientation) is driven by the pump laser pulse, but it is also coupled with the translation motion (flow) of the fluid through viscosity. Thus, with a finite pump beam, a rigorous theoretical calculation would require the solution of a set of coupled three-dimensional nonlinear partial differential equations for the angle of... 

The time dependence of the molecular reorientation following the usual Ericksen-Leslie approach becomes [11]... 

Within the framework of the Ericksen, Leslie, and Parodi theory one can obtain the torque acting on a sheared molecule. 

Using again the theory of Ericksen, Leslie, and Parodi together with the momentum conservation we obtain the differential equations that govern the steady state of the system... 

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