The Frank-Oseen Elastic Energy

The static theory of nematic and cholesteric liquid crystals employs the unit vector field n, frequently referred to as the director, to describe the mean molecular alignment at a point x in a given sample volume V, Hence 

This alignment displays a certain elasticity and it is known that an initial uniform alignment of a nematic liquid crystal commonly returns after the removal of any disturbing influences. It is therefore assumed that there is a free energy density, also called the free energy integrand, associated with distortions of the anisotropic axis of the form 

The free energy per unit volume must also be the same when described in any two frames of reference, that is, it must be frame-indifferent. This means that the energy density must be invariant to arbitrary superposed rigid body rotations and consequently we require 

We develop the approach adopted by Frank [91]. Let n be a unit vector representing the preferred orientation of a uniaxial liquid crystal at any point x and assume that it varies slowly with position. The sign of n has no physical significance in most cases. However, for molecules with permanent dipole moments this may not be the case and then the sign of n becomes important, but this will not be considered here. We introduce a local system of Cartesian coordinates x, z with z parallel to 

The curvature strains can also be obtained by considering a Taylor series expansion for the components of n about the origin. Firstly, because n n = 1, it follows that 

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

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... 

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. 

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]. 

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 ... 

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... 

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... 

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]... 

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)). 

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... 

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... 

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