Oseen nematics

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. 

Fig. 7.2. Oseen s three elementary elastic deformations do not have a centre of symmetry. Therefore one might think they should be accompanied by an induced polarization analogous to the piezo-effect in solids. This is partly true. A splay violates the invariance under n —> — n, hence the P vector is in the direction of n. The invariance holds for bend and twist but in a bend deformation P is perpendicular to n. This is not possible in a twist because this deformation has a two-fold rotation axis Xn that inverts any PXn. Thus a nematic has only two flexoelectric coefficients.

We can imagine a cholesteric as a smck of nematic quasi-layers of molecular thickness a with the director slightly turned by ( ) from one layer to the next one. In fact it is Oseen model [18]. Such a structure is, to some extent, similar to lamellar phase. Indeed, the quasi-nematic layers behave like smectic layers in formation of defects, in flow experiments, etc. Then, according to the Landau-Peierls theorem, the fluctuations of molecular positions in the direction of the helical axis blur the one-dimensional, long-range, positional (smectic A phase like) helical order but in reality the corresponding scale for this effect is astronomic. 

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 theory of wave propagation in cholesterics was first developed by Oseen and later by de Vries. Their theories were based on the assumption that the cholesteric structure is simply a twisted version of the nematic. The director rotates in space around an axis forming thereby a helical structure with a pitch p. 

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... 

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 ... 

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 first two terms are the Landau part, Eq. (17). Because of the nematic anisotropy, the gradient terms exhibit anisotropic coefficients (Cn s Cx) along directions parallel and perpendicular to the director n. With the notation = 3/3z and = (3/3jc, 3/3y) and at lowest relevant order in (5nj ), these gradients have the form Eq. (20). The last three terms are the usual Frank-Oseen elastic energy of the nematic [21]. 

A quite different model for nematic dif-fusivities, based on Oseen s hydrodynamic theory of isotropic liquids, was elaborated by Franklin [14] it describes the diffusivity components in terms of the five Leslie viscosities a, to 05, a scalar friction constant... 

The molecular theory is similar to Cauchy s description of the elastic theory of solids [1] and utilizes additive local molecular pair interactions to describe elasticity. The latter approach was taken by Oseen [2], who was the first to establish an elastic theory of anisotropic fluids. Oseen assumed short-range intermolecular forces to be the reason for the elastic properties, and he derived eight elastic constants in the expression for the elastic free energy density of uniaxial nematic phases. Finally, he retained only five of them, which enter the Euler-Lagrange equations describing equilibrium deformation states of the nematic mesophase, and omitted the other three. 

The elastic constant introduced by Nehring and Saupe [5] is the only coefficient in the second-order elastic theory of ordinary nematics that involves explicitly second derivatives of the director field. Apart from its practical consequences, is of some theoretical significance because of the relations K i=Kxi+2Ki and Kj =Kj -2Ki2 between the Oseen-Neh-ring-Saupe coefficients and... 

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... 

These optical properties of chiral nematic materials have all been observed experimentally. There have been quite extensive theoretical studies carried out by Mauguin [60], Oseen [61], and de Vries [26] to explain how these properties arise from the he-licoidal structure. Kats [62] and Nityanan-da [63] have derived exact wave equations to explain the propagation of light along the optic axis and Friedel [32] has reviewed the main textures observed with chiral nematics. We will outline the important elements of these studies in the next sections (Secs. 2.2.1.1 -2.2.1.3). In Sec. 2.2.1.4 we will consider how the helicoidal pitch, and... 

The textures for achiral nematics have been treated in great detail by Nehring and Saupe [68], following the pioneering theoretical work of Oseen [61] and Frank [65], and more recently comprehensive reviews have been presented [69, 70]. We are now in a position to consider the more complex case of chiral nematics. 

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... 

We use the Oseen-Frank elastic energy expression [Eq. (96)] for a nematic medium as a starting point. Now, according to our assumption, the medium is chiral, and an ever so slight chiral addition to a nematic by symmetry transforms the twist term according to [111]... 

Inserting this in Eq. (216) we find that the Lifshitz invariant (which has a composition rule slightly reminiscent of angular momentum, cf. Lj( = xpy-ypx) in the cholesteric case has the value equal to q, the wave vector. In fact we can gain some familiarity with this invariant by starting from an expression we know quite well, the Oseen-Frank expression for the elastic free energy Eq. (96). Because of its symmetry, this expression cannot describe the cholesteric state of a nematic which lacks reflection symmetry and where the twisted state represents the lowest energy. Now, if there is a constant twist with wave vector q, the value of n Fxn in the K22 term equals-q. The expression Eq. (96) therefore has to be renormalized to... 

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

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). 

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... 

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. 

Fundamental theories of diffusion for low-molecular weight liquid crystals in the nematic phase have been studied by Franklin [103-105] based on Oseen-Kirkwood hydrodynamic theory for isotropic liquids. Further theories of diffusion for low-molecular weight liquid crystals have been developed. These theories explained partially the experimental data on Dy and D. Chu and Moroi [106], and Leadbetter [107] have obtained that the anisotropy ratio of the diffusion coefficients, Dy/Dj, for low-molecular weight liquid crystals are expressed by [2y(l - S) + 2S -b 1]/[7(S + 2) -b 1 — S], where y = Ttd/Al in which I is the length and d of the diameter of the... 

The effect of surface elastic constants on the nematic director configurations is of basic interest for the elastic theory of liquid crystals and plays a critical role in those device applications where the nematic is confined to a curved geometry. The saddle-splay surface elastic constant, K24, and the splay-bend surface elastic constant, K13, defied measurement for more than sixty years, since the pioneering work of Oseen, who made the first steps toward the elastic theory of liquid crystals. 

The Oseen-ZOcher-Frank equation [13] for the free energy density F allows a good understanding of chiral nematic phases [1,2] ... 

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]. 

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. 

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... 

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