Ericksen-Leslie theory, nematics

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 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. ... 

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]. 

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... 

When De 1 and molecular elasticity is negligible, the flow properties of polymeric nematics can, in principle, be described by the Leslie-Ericksen equations (see Section 10.2.3). However, at moderate and high De, the Leslie-Ericksen continuum theory fails, and a molecular theory is required to describe the effect of flow on the distribution of molecular orientations. 

At low enough shear rates, polymeric nematics ought to obey the same Leslie-Ericksen continuum theory that describes so well the behavior of small-molecule nematics. The main difference is that polymers have a much higher molecular aspect ratio than do small molecules, which leads to greater inequalities in the the numerical values of the various viscosities and Frank constants and to much higher viscosities. 

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 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). 

The constitutive hydrodynamic equations for uniaxial nematic calamitic and nematic discotic liquid crystals are identical. In comparison to nematic phases the hydro-dynamic theory of smectic phases and its experimental verification is by far less elaborated. Martin et al. [17] have developed a hydrodynamic theory (MPP theory) covering all smectic phases but only for small deformations of the director and the smectic layers, respectively. The theories of Schiller [18] and Leslie et al. [19, 20] for SmC-phases are direct continuations of the theory of Leslie and Ericksen for nematic phases. The Leslie theory is still valid in the case of deformations of the smectic layers and the director alignment whereas the theory of Schiller assumes undeformed layers. The discussion of smectic phases will be restricted to some flow phenomena observed in SmA, SmC, and SmC phases. 

The macroscopic theory that takes into accoimt the effect of the orientation order was developed by Ericksen, Leslie and Parodi, and usually is referred as ELP theory. A microscopic theory based on correlation functions, which then were "translated" to macroscopic terms and extended to other mesomorphic phases, was developed by the Harvard group. Although usually tire ELP flieory is accepted, it seems that the two approaches are equivalent. A continuum theory of biaxial nematics was developed by Saupe, who followed the description we give with (4.1)-(4.8). In the uniaxial situation, they reproduce the Leslie-Ericksen and Harvard theories. 

The general theory of shear flow normal to the helical axis has been discussed by Leslie. This basically uses concepts similar to those described by the Ericksen-Leslie-Parodi theory of nematics. The main difference is that the twist term of the deformation free energy will be ... 

Much research in the last few decades focused on the simulation of LCPs for various processes. Suitable rheological constitutive equations are required for this simulation. Leslie-Ericksen (LE) theory describes the flow behaviour and molecular orientation of many LCPs. LE model is limited to low shear rates and weak molecular distortions. But at high shear rate, the rate of molecular distortions is too fast. Doi and Edwards developed their model to describe the complex dynamics of macromolecules at high shear rate (Doi and Edwards 1978). Doi theory is applicable for lyotropic LCPs of small and moderate concentrations. Due to the complex nature of Doi theory, it is also challenging for simulation. Leonov s continuum theory of weak viscoelastic nematodynamics, developed on the basis of thermodynamics and constitutive relations, consider the nematic viscoelasticity, deformation of molecules as well as evolution of director. 

It is convenient at this point to summarise the Ericksen-Leslie dynamic equations for nematics in the incompressible isothermal theory when the director inertial term is neglected. These are the most frequently used forms of the equations and we state them in the notation introduced in the previous Sections. They consist of the constraints... 

F.M. Leslie, Theory of Flow Phenomena in Nematic Liquid Crystals, in Theory and Applications of Liquid Crystals, J.L. Ericksen and D. Kinderlehrer (Eds.), 235-254, Springer-Verlag, New York, 1987. 

Chapter 1 gives a brief introduction to some of the elementary aspects and descriptions of liquid crystals and helps to set the scene for later Chapters. The static theory of nematic liquid crystals is developed in Chapter 2 while Chapter 3 goes on to discuss some applications of this theory which have particular physical relevance. The dynamics of nematics, leading to the celebrated Ericksen-Leslie dynamic equations, are fully derived in Chapter 4, with Chapter 5 providing some detailed accounts of applications of this dynamic theory. 

The rigid nature of the mesophase pitch molecules creates a strong relationship between flow and orientation. In this regard, mesophase pitch may be considered to be a discotic nematic liquid crystal. The flow behavior of liquid crystals of the nematic type has been described by a continuum theory proposed by Leslie [36] and Ericksen [37]. 

Leslie, F. M., "Theory of Flow in Nematic Liquid Crystals, The Breadth and Depth of Continuum Mechanics—A Collection of Papers Dedicated To J. L. Ericksen, C. M. Da-fermos, D. D. Joseph, andF. M. Leslie, Eds., Springer-Verlag, Berlin, 1986. 

In a weak flow field, Eq. (64) can be rewritten in a form similar to that for the direct n appearing in Leslie and Ericksen s phenomenological theory [160—163] for nematic systems. Thus, we have... 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

Inspired by a mesoscopic theory for mixtures of immiscible liquids (Doi and Ohta 1991 see also Section 9.3.3), Larson and Doi (1991) have derived mesoscopic equations for polydomain nematics by multiplying the Leslie-Ericksen equation (10-13) by the director... 

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