The Classical Freedericksz Transitions in Nematics

The three classical Freedericksz transitions will be considered in what are commonly called the splay, twist and bend geometries. Full details will be given for the splay geometry, with the corresponding results for the twist and bend geometries being stated since their analysis is analogously similar. The Section ends with a qualitative discussion for the one-constant approximation. 

Consider a nematic liquid crystal sample confined between two parallel glass plates a distance d apart and suppose that, in the geometry shown in Fig. 3.6, the director 

Inserting n and H into the Frank elastic energy (2.49) and magnetic energy (2.107) gives, with = d/dz as before, 

The relevant equilibrium equation is obtained from equation (2.221) and is 

On the interval 0 z 6 is expected to increase monotonically from zero to 

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