Leslie-Ericksen continuum theory

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. 

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 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. 

There are two basic theories to describe the LC state, the continuum theory mainly proposed by Oseen, Zocher, and Frank and the swarm theory supported by researchers such as Bose, Bom, Omstein, Maier, and Saupe [10,11]. The continuum theory models the liquid crystal as an anisotropic elastic medium with properties varying as a function of position. The swarm theory emphasizes molecular interactions and interprets the LC state as the result of a statistically driven thermodynamic equilibrium. In the recent work of de Gennes, Leslie and Ericksen, LC theories integrate aspects of both the continuum theory and the swarm theory [11]. 

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]. 

Liquid Crystal Continuum Theory (Leslie/Ericksen)... 

None of these phenomena - magnetic field or surface orientation dependence, tube diameter dependence, and the shear rate dependence at low shear rates - are observed with Isotropic fluids under comparable conditions. They are all predicted very satisfactorily by the Lesiie-Ericksen (L-E) continuum theory of the mechanics of liquid crystals. It will not be attempted here to give more than a cursory account, copied from Reference 2, of the theory complete descriptions are given in the references cited in Reference 3, especially Leslie s review article. 

Tendencies to instability in nature have been interpreted in various ways in continuum theory. We recall that many substances exhibit several phase transitions as, for example, their temperature is increased. A material initially described as a rigid solid may pass through smectic and nematic liquid crystal phases prior to behaving like an isotropic liquid. In a liquid crystal polymer, the concentration of solvent sometimes has the role of temperature. In the liquid crystal phase, the orientation of the molecules in terms of the optical axis may contribute to the response of this "fluid" to external fields. It is sometimes called an internal variable". The traditional field equations for an isotropic liquid are replaced by a more elaborate collection derived on the basis of continuum theory, (Ericksen [10], Leslie, [16], Hissbrun [H]). 

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 Leslie-Ericksen theory for flow of nematics is a continuum theory which considers the coupling between velocity field and director field. Details about this important theory are presented in Vertogen and de Jeu (1988). 

This continuum theory models many static and dynamic phenomena in nematic liquid crystals rather well, and various accounts of both the theory and its applications are available in the books by de Gennes and Frost [8], Chandrasekhar [9], Blinov [10] and Virga [11], and also in the reviews by Stephen and Straley [12], Ericksen [13], Jenkins [14] and Leslie [15]. Given this suc-... 

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. 

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. 

Oseen [1] and Frank [2] far before the development of LCD technology. The dynamic continuum theory of nematics, which is frequently called the nematodynamics, was developed by Ericksen [3] and Leslie [4] (hereafter referred to as E-L theory) based on the classical mechanics just in time for the upsurge of LCD technology. In conjunction with the electrodynamics of continuous media, the static and dynamic continuum mechanics of Oseen-Erank and E-L theory provided theoretical tools to analyze quantitatively key phenomena, e.g., Freedericksz transition of various configurations and associated optical switching characteristics. For the details of E-L theory [5-7] and its development [9,10], please refer to the articles cited. 

In general, the classical Fredericks transition in nematics can be fairly well-explained using continuum theory of nematic liquid crystals developed by Frank, Ericksen and Leslie. Before we present a detailed analysis on the optical Fredericks transition, which couples the interaction between the applied electromagnetic field of a light wave and the orientation of hquid crystals, we would like to briefly review the classical results (de Geimes and Frost 1993 Virga 1994 Stewart 2004). Many of our ideas here are borrowed from Stewart (2004). 

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... 

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