The Frank-Oseen Energy

W again denoting the Frank-Oseen energy and A an arbitrary scalar. 

Some 25 years ago, Nehring and Saupe [16] proposed that one should add terms linear in second gradients of the director to the Frank-Oseen energy, this ultimately entailing the inclusion of a single additional term, namely... 

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]. 

Worked Example 10.1 shows how to calculate the Frank-Oseen free energy and use it to predict the response of a liquid crystal to a magnetic or electric field. Such calculations are used to design practical liquid-crystal display devices. They also can be used to determine the values of the elastic constants. 

An elastic torque can be derived form the Frank-Oseen elastic energy to give... 

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... 

Topologically, it turns out that the helical structure of the cholesteric cannot be deformed continuously to produce a cubic lattice without creating defects. Thus BP I and BP II are unique examples in nature of a regular three-dimensional lattice composed of disclination lines. Possible unit cells of such a disclination network, arrived at by minimizing the Oseen-Frank free energy, are shown in fig. 4.8.3. The tubes in the diagram represent disclination lines, whose cores are supposed to consist of isotropic (liquid) material. Precisely which of these configurations represents the true situation is a matter for further study. 

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 note immediately that, when H is strictly perpendicular to n, the magnetic field has no effect. Only fluctuations 8n in n allow the field to act upon the orientation of the director. The elastic energy can be simplified by observing that, for each case represented in Fig. 9.6, the deformation is associated with a single Frank-Oseen constant Ki, and that it depends only on z. Hence we can write... 

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]. 

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... 

We have seen that the soft elastic energy of a smectic C material can be derived from the Oseen-Frank energy of nemahc liquid crystals (see (4.56)). 

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