Frank-Oseen Theory

Here V n and V x n are the divergence and the curl of n. The three contributions to Wd are associated with the three independent modes of distortion splay, twist, and bend, depicted in Fig, 10-6. Terms of higher order than quadratic in Vn are only required if spatial distortions become severe. The Frank constants K, K2, and are of the order u /a, 

Analyses are simplified by taking all three constants to be equal (the one-constant approximation) K = K] = K2 = K3. It can then be shown that 

Here T denotes transpose, and has been expressed in terms of both Gibbs and Einstein s 

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The LE theory is rather complex since it contains both viscous and elastic stresses. It can best be understood by considering viscous and elastic effects separately. If elastic effects are neglected, the LE equations reduce to Ericksens transversely isotropic fluidy while in the absence of flow the elastic stresses are just those of the Frank-Oseen theory (discussed below in Section 10.2.2). ... 

A further recent innovation is due to Ericksen [17] who proposes an extension to the Frank-Oseen theory in order to improve solutions modelling defects. To this end he incorporates some variation in the degree of alignment or the order parameter, and therefore proposes an energy of the form... 

One optical feature of helicoidal structures is the ability to rotate the plane of incident polarized light. Since most of the characteristic optical properties of chiral liquid crystals result from the helicoidal structure, it is necessary to understand the origin of the chiral interactions responsible for the twisted structures. The continuum theory of liquid crystals is based on the Frank-Oseen approach to curvature elasticity in anisotropic fluids. It is assumed that the free energy is a quadratic function of curvature elastic strain, and for positive elastic constants the equilibrium state in the absence of surface or external forces is one of zero deformation with a uniform, parallel director. If a term linear in the twist strain is permitted, then spontaneously twisted structures can result, characterized by a pitch p, or wave-vector q=27tp i, where i is the axis of the helicoidal structure. For the simplest case of a nematic, the twist elastic free energy density can be written as ... 

A theoretical relation between the nematic elastic constants and the order parameter, without the need for a molecular interpretation, can be established by a Landau-de Gennes expansion of the free energy and comparison with the Frank-Oseen elastic energy expression. While the Frank theory describes the free energy in terms of derivatives of the director field in terms of symmetries and completely disregards the nematic order parameter. The Landau-de Gennes expansion expresses the free energy in terms of the tensor order parameter 0,-, and its derivatives (see e.g. [287,288]). For uniaxial nematics, this spatially dependent tensor order parameter is... 

Frank theory, nematics 60 Frank-Oseen energy 27 Frederiks threshold... 

In all of the present theories about the excitation of nematic or cholesteric liquids by an electric field, the mesomorphic material is treated as a continuous elastic anisotropic medium. The Oseen -Frank elastic theory is used to describe the interaction between the applied field and the fluid. The application of an electric field causes the liquid crystal to deform. For a material with a positive dielectric anisotropy, Ae = > 0, the director aligns in the direction of... 

Note 3 The names of Oseen, Zocher, and Frank are associated with the development of the theory for the elastic behaviour of nematics and so the elastic constants may also be described as the Oseen-Zocher-Frank constants, although the term Frank constants is frequently used. 

The tendency of LCs to resist and recover from distortion to their orientation field bears clear analogy to the tendency of elastic solids to resist and recover from distortion of their shape (strain). Based on this idea, Oseen, Zocher, and Frank established a linear theory for the distortional elasticity of LCs. Ericksen incorporated this into hydrostatic and hydrodynamic theories for nematics, which were further augmented by Leslie with constitutive equations. The Leslie-Ericksen theory has been the most widely used LC flow theory to date. 

The elasticity theory of liquid crystals was proposed by Oseen (1933) and Zocher (1933), and then modified by Frank into the form that has since... 

Distortions and defects can be interpreted in terms of the continuum theory through equations derived from the expressions of the elastic energy and the imposed boundary conditions. Solutions are known in certain simple situations. Oseen [35] has found configurations, named disinclinations by Frank [33], or disclinations today, which are solutions of this problem for planar samples in which the director n is confined to... 

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]. 

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... 

Oseen s elastic theory was developed further by Zocher [3] and re-examined by Frank [4] in a phenomenological approach. 

The foundations of continuum theory were first established by Oseen [61] and Zocher [107] and significantly developed by Frank [65], who introduced the concept of curvature elasticity. Erickson [17, 18] and Leslie [15, 16] then formulated the general laws and constitutive equations describing the mechanical behavior of the nematic and chiral nematic phases. 

As discussed in Sec. 2.2.2.1, the foundations of the continuum model were laid by Oseen [61] and Zocher [107] some seventy years ago, and this model was reexamined by Frank [65], who introduced the concept of curvature elasticity to describe the equilibrium free energy. This theory is used, to this day, to determine splay, twist, and bend distortions in nematic materials. The dynamic models or how the director field behaves in changing from one equilibrium state to another have taken much longer to evolve. This is primarily due to the interdependency of the director it (r, t) and v (r, /) fields, which in the case of chiral nematics is made much more complex due to the long-range, spiraling structural correlations. The most widely used dynamic theory for chiral... 

One well-known characteristic feature of nematic liquid crystals is the thread-like texture that can be observed with a polarizing microscope. The name nematic, derived from the Greek word "thread," reflects that feature. By examining the thin and thick thread-like structures in nematic liquid crystals, Otto Lehman i and Georges FriedeF deduced that this phase involves long-range orientational order. The first step to the interpretation of the threads as disclinations of the director field has been made by Oseen. Later Frank " derived Oseen s theory of curvature elasticity on a more general basis and presented it in a simpler form (see Appendix C.1). 

The phenomenological (continuous) theories originate with the theory of Oseen [44] and Frank [45] and were most completely expressed in the studies of Ericksen [46] and Leslie [47]. In these theories, the liquid crystal is considered an anisotropic (oriented) liquid. Tensor values characterizing the nonequivalence of the rheological properties of the medium in different directions are introduced in the rheological equations of state of anisotropic liquids. The most common form of the correlation between the stress field 0 and the field of the rates of deformation Y/jt linearly viscous anisotropic liquid is [48]... 

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. 

The presently accepted continuum theory for liquid crystals has its origins going back to at least the work of Oseen [214, 215], from 1925 onwards, and Zocher [286] in 1927. Oseen derived a static version of the continuum theory for nematics which was to be of instrumental importance, especially when the static theory was further developed and formulated more directly by Frank [91] in 1958. This static theory, introduced in Chapter 2, is based upon the director n and its possible distortions. 

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