Continuum Theory and Free Energy

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. 

In the continuum theory of liquid crystals, the free energy density (per unit volume) is derived for infinitesimal elastic deformations of the continuum and characterized by changes in the director. To do this we introduce a local right-handed coordinate x, y, z) system with (z) at the origin parallel to the unit vector n (r) and x and y at right angles to each other in a plane perpendicular to z. We may then expand n (r) in a Taylor series in powers of x, y, z, such that the infinitesimal change in n (r) varies only slowly with position. In which case 

Since we are considering infinitesimal changes, we may ignore 0 (r ) terms and partial derivatives of are zero. The components of strain are then (see Fig. 27) 

Free energy considerations allow us to postulate [59] for an incompressible fluid that the free energy F of the liquid crystal in any particular configuration, relative to that in a state of uniform orientation, is the volume integral off, a free energy density, which is a quadratic function of the six differential coefficients which measure the curvature a,- 

An arbitrary choice of coordinate system. We could therefore redefine a permissible system x, y and z with new curvature components kj. As the free energy density,/, must be as before, this restricts the moduli k,- and ky. 

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