Oseen-Frank expression

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... 

In other words, both twist and bend distortions are absent, leaving only the splay term in the Oseen-Frank free energy expression (3.3.7). It is seen from fig. 5.3.1, that by merely bending or corrugating the layers a splay deformation can be readily achieved without affecting the layer thickness. 

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]... 

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... 

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... 

Oseen was also the first to realize the importance of the n- -n invariance [110], which he used to derive Eq. (96). This expression was rederived thirty years later by Frank in a very influential paper [111],... 

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]... 

An accessible review of the Frank-Oseen energy for the bulk expressions is available in the book by Collings and Hird [25] while more detailed, yet concise, comments are available in the recent reviews of Leslie [26,27] and of Dunmur and Toriyama [28]. Fuller discussions and examples of applications are to be found in the reviews by Stephen and Straley [8] and Ericksen [29], and the books by de Gennes and Frost [5] and Chandrasekhar [6]. 

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