Orientational Order, Elasticity and Singularities

Our experiments have shown us that nematic phases are anisotropic liquid crystals, that is, their properties are not the same in every spatial direction. The long thin molecules are randomly distributed but tend to line up parallel to one another, as shown in Fig. 9.0b. However, at any given instant, each molecule is inclined at a small angle a to the average direction over all 

Consider first a small deviation 8n = n — no of the director n from the reference direction no, resulting from a displacement 

We have 8n no = 0. Choosing the z direction to be along no, this means that to first order, 

The first, second and fourth terms correspond to the three fundamental deformations mentioned earlier. 

The third term describes a more complex deformation called saddle-splay. Indeed, it is given by a symmetric real matrix with zero trace. It can therefore be diagonalised in some orthonormal frame with axes Xq, yo, relative to which it takes the form  

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