Nematic liquid crystal continuum theory

F.M. Leslie, Continuum theory for nematic liquid crystals, Continuum Mech, Thermodyn., 4, 167-175 (1992). 

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. 

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]. 

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 initial research on electro-optic phenomena in side-chain polymer liquid crystals concentrated on systems that exhibited nematic phases so that a ready comparison could be made with low molar mass mesogens. Such measurements have established that electro-optic devices are feasible and have allowed elastic constants to be deduced from applications of the continuum theory. This theory, originally derived for low molar mass nematic liquid crystals, defines a relationship for the free energy density F in terms of the elastic constants (/ ) and the director n such that ... 

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 superiority of using lasers for material studies often lies in its spatial and temporal flexibilities, that is, the material can selectively excited and probed in space and time. These qualities may allow us to elucidate fundamental material properties not accessible to conventional techniques. The location, dimension, direction, and duration of the material excitation can be readily controlled through adjustment of the beam spot, direction, polarization, and pulse width of the exciting laser field. The flexibilities can be further enhanced when two or more light waves are used to induce excitations. Such a technique, however, has not yet been fully explored in liquid-crystal research. Although the recent studies of optical-field-induced molecular reorientation in nematic liquid-crystal films have demonstrated the ability of the technique to resolve spatial variation of excitations, corresponding transient phenomena induced by pulsed optical fields have not yet been reported in the literature. Because of the possibility of using lasers to induce excitations on a very short time scale, such studies could provide rare opportunities to test the applicability of the continuum theory in the extreme cases. 

The elastic continuum theory is based on the assumption that at each point within the liquid crystal a preferential direction for the molecular orientation is given which is described by a unit vector L, and which varies continuously from place to place — except for a few singular lines or points. Any distortion of the undisturbed state requires a certain amount of energy since elastic torques attempt to maintain the original configuration. The elastic energy density of a deformed nematic liquid crystal is given by... 

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-... 

Continuum theory generally employs a unit vector field n(x) to describe the alignment of the anisotropic axis in nematic liquid crystals, this essentially ignoring variations in degrees of alignment which appear to be unimportant in many macroscopic effects. This unit vector field is frequently referred to as a director. In addition, following Oseen [1] and Frank [4], it commonly assumes the existence of a stored energy density W such that at any point... 

An important aspect of the macroscopic structure of liquid crystals is their mechanical stability, which is described in terms of elastic properties. In the absence of flow, ordinary liquids cannot support a shear stress, while solids will support compressional, shear and torsional stresses. As might be expected the elastic properties of liquid crystals are intermediate between those of liquids and solids, and depend on the symmetry and phase type. Thus smectic phases with translational order in one direction will have elastic properties similar to those of a solid along that direction, and as the translational order of mesophases increases, so their mechanical properties become more solid-like. The development of the so-called continuum theory for nematic liquid crystals is recorded in a number of publications by Oseen [ 1 ], Frank [2], de Gennes and Frost [3] and Vertogen and de Jeu [4] extensions of the theory to smectic [5] and columnar phases [6] have also been developed. In this section it is intended to give an introduction to elasticity that we hope will make more detailed accounts accessible the importance of elastic properties in determining the... 

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. 

Among the characteristic physical properties of liquid crystals, what are of critical importance to display devices (LCDs) are those of macroscopic spatiotemporal scale there, the theories of liquid crystals as continuous media play essential roles. The basis of static continuum mechanics of nematic liquid crystals was established by... 

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). 

Stark H (2001) Physics of colloidal dispersions in nematic liquid crystals. Phys Rep 351 387-474 Stewart IW (2004) The static and dynamic continuum theory of liquid crystals a mathranatical introduction. Taylor Francis, London... 

Continuum theory for nematic liquid crystals has its origins in the 1920s in the work of Oseen [1] and Zocher [2], who largely developed the static theory. The first to attempt the formulation of a dynamic theory was Anzelius [3], who was a student of Oseen, but an acceptable version had to await developments in non-linear continuum mechanics many years later, as well as further experimental studies by Zwetkoff [4] and Miesowicz [5]. A full account of the early development of dynamic theory for nematics can be found in a paper by Carlsson and Leslie [6]. 

Certain defects in nematic liquid crystals were discussed in a mathematical way by Oseen [215] in 1933 and later by Prank [91] in 1958. These defects, and others, are described in some detail in Section 3.8. However, not all defects can be adequately described by the classical continuum theory mentioned above, and this led Ericksen [82] to return to the equilibrium theory of nematic liquid crystals in... 

The static theory of nematic liquid crystals is introduced in this Chapter. Static continuum theory involves two major steps. The first step is to construct an energy based upon possible distortions of the director n, introduced in Chapter 1. The second step is to minimise this energy in some sense, and this leads to equilibrium equations involving n and its derivatives. The solutions of these differential equations yield possible equilibrium orientations for n it is these alignments of n that are the ultimate goal of static continuum theory since they indicate the director alignment within a sample of liquid crystal. The solutions with the least energy are interpreted as the physically relevant ones. 

G. Barbero and L.R. Evangelista, An Elementary Course on the Continuum Theory for Nematic Liquid Crystals, World Scientific, Singapore, 2001. 

F.M. Leslie, J.S. Laverty and T. Carlsson, Continuum Theory for Biaxial Nematic Liquid Crystals, Q. Jl. Mech. Appl. Math. 45, 595-606 (1992). 

H.C. Tseng, D.L. Silver and B.A. Finlayson, Application of the Continuum Theory to Nematic Liquid Crystals, Phys. Fluids, 15, 1213r-1222 (1972). 

The results and applications in Chapters 2 to 5 for nematic liquid crystals are given in fairly full mathematical detail. It has been my experience that the stumbling block for many people comes at the first attempts at the actual calculations here I will reveal many details and more explanation than is usually given in articles and common texts, in the hope that readers will gain confidence in how to apply the main results from continuum theory to practical problems. These Chapters contain extensive derivations of the static and dynamic nematic theory and applications. Chapter 6, on the other hand, does not give as many detailed computations as those presented in the earlier Chapters it is my intention that it introduces the reader to a continuum theory of smectic C liquid crystals and it is probably written more in the style of an introductory review. This is partly because some of the calculations are similar for both nematic and smectic C materials, but with different physical parameters and some different physical interpretations. However, despite some of these similarities, smectic liquid crystals have some uniquely different mathematical problems, and these can only be touched upon within the remit of a book such as this. 

In nematic phase, the liquid crystal director it is uniform in space in the ground state. In reality, the liquid crystal director it may vary spatially because of confinements or external fields. This spatial variation of the director, called the deformation of the direetor, eosts energy. When the variation occurs over a distance much larger than the moleeular size, the orientational order parameter does not change, and the deformation ean be deseribed by a continuum theory in analogue to the classic elastic theory of a solid. The elastie energy is proportional to the square of the spatial variation rate. 

In addition to the commonly used kinematic and dynamic variables of continuum theories, the L-E theory contains a unit vector, n, called the "director", to describe the orientation of the liquid crystal. There is an elastic energy, W, associated with spatial variations of the director. For nematics, this is expressed as... 

Theoretical treatments of liquid crystals such as nematics have proved a great challenge since the early models by Onsager and the influential theory of Maier and Saupe [34] mentioned before. Many people have worked on the problems involved and on the development of the continuum theory, the statistical mechanical approaches of the mean field theory and the role of repulsive, as well as attractive forces. The contributions of many theoreticians, physical scientists, and mathematicians over the years has been great - notably of de Gennes (for example, the Landau-de Gennes theory of phase transitions), McMillan (the nematic-smectic A transition), Leslie (viscosity coefficients, flow, and elasticity). Cotter (hard rod models), Luckhurst (extensions of the Maier-Saupe theory and the role of flexibility in real molecules), and Chandrasekhar, Madhusudana, and Shashidhar (pre-transitional effects and near-neighbor correlations), to mention but some. The devel-... 

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