Ericksen-Leslie dynamic equations

The Ericksen-Leslie Dynamic Equations and note that this implies... 

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... 

The linearised version of the Ericksen-Leslie dynamic equations coincides with that discussed by Martin, Parodi and Pershan [192], as mentioned by Leslie [168], with suitable reorganisation of the terms in the equations. 

The constraints in equation (4.118) are clearly satisfied. The governing Ericksen-Leslie dynamic equations (4.119) and (4.120) become, respectively,... 

The Ericksen-Leslie dynamic equations therefore finally reduce to the two equations... 

Before proceeding to investigate typical shear flow behaviour in the following subsections, we examine and comment upon a particularly simple shear flow solution to the usual Ericksen-Leslie dynamic equations. Consider a shear flow in the x-direction between two flat plates in which the director is uniformly aligned perpendicular to the plane of shear, with the director fixed parallel to the bounding plates at y = 0 and y = d. Suppose also that there is no external body force F or generalised body force G. Seeking solutions in Cartesian coordinates of the form... 

In these circumstances, motivated by the set-up in Fig. 5.7, it is appropriate in this basic investigation to seek solutions to the Ericksen-Leslie dynamic equations of the form... 

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 six viscosities in equation (4.161) to (4.164) and (4.166) can now all be given in terms of the five independent Leslie viscosities arising in the Ericksen-Leslie dynamic theory, assuming the widely accepted Parodi relation (4.96) holds. Hence only five of these viscosities are needed to form a canonical set of viscosities. 

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]. 

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... 

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). 

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. 

For finite wavelengths, the collective dynamics of bulk nematics can be described within the hydrodynamic equations of motion introduced by Ericksen [4-8] and Leslie [9-11]. A number of alternate formulations of hydrodynamics [12-18] leads essentially to the equivalent results [19]. The spectrum of the eigenmodes is composed of one branch of propagating acoustic waves and of two pairs of overdamped, nonpropagating modes. These can be further separated into a low- and high-frequency branches. The branch of slow modes corresponds to slow collective orientational relaxations of elastically deformed nematic structure, whereas the fast modes correspond to overdamped shear waves, which are similar to the shear wave modes in ordinary liquids. 

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. 

The derivation of the equilibrium equations for SmC liquid crystals parallels that outlined in Section 2.4 for nematic and cholesteric liquid crystals, this approach being based on work by Ericksen [73, 74]. The energy density will be described in terms of the vectors a and c, and the equilibrium equations and static theory will be phrased in this formulation these vectors turn out to be particularly suitable for the mathematical description of statics and dynamics. We assume that the variation of the total energy at equihbrium satisfies a principle of virtual work for a given volume V of SmC liquid crystal of the form postulated by Leslie, Stewart and Nakagawa [173]... 

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