Elasticity and anchoring

Let us forget preferential wetting for the moment and introduce the usual formulae for nematic elasticity and anchoring energy in the framework of the continuous description  

The complete equations, where both n(r) and 5(r) are variable, are very complicated. Nevertheless, in the presence of antagonist anchorings, one may reasonably assume that the main contribution to the free energy comes from the spatial derivatives of/i(r) and treat S(r) as being a constant S. Then the spatial gradients in the free energy appear as the usual nematic elasticity  

The three (positive) elastic constants Kn (splay), K22 (twist), and K33 (bend) are associated to the three principal deformations. In the surface term, fs is the contribution of the two anchorings, k is the unit vector normal to the surface and directed outward, K13 is the splay-bend constant, and K24 is the saddle-splay constant. The two last surface terms play only for thin films the mere existence of the splay-bend constant K13 is a matter of debate. In the framework of Landau-de Gennes analysis, = K33 and the elastic 

At the free interface, Wh is the homeotropic anchoring energy and 9 the actual angle between n and the normal to the film. On the substrate, Wy is the planar anchoring energy and 9y the actual angle between n and the normal. 

Let us now discuss more specifically experiments on extended nematic films, either spin-coated on silicon wafers or deposited on water or glycerol. Usually, a solution of the liquid cr5 stal in hexane is deposited, allowing a fast spreading on the substrate, followed by a rapid evaporation of hexane. The first experiments with 5CB on wafers reported before have been completed by a systematic investigation of the series of n-cyanobiphenyls on water and glycerol in the nematic range. The aim of the study is to define a common scheme of behavior for the nCB series on the three isotropic substrates, as a base for the theoretical analysis. Specific deviations from the common scheme must also be understood. 

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Although stripe patterns are controlled by a competition between elasticity and anchoring, which both decrease strongly close to the N1 transition, the main effect on ) is the increase of the UB and seemingly a decrease of D. The curve itself depends only weakly on temperature. 

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