Relationship between elasticity and orientational order

5 Relationship between elasticity and orientational order As remarked in chapter I, a uniformly oriented film of nematic liquid crystal may be prepared by prior treatment of the surfaces with which it is in contact. If the preferred orientation imposed by the surfaces is perturbed, let us say by a magnetic field, a curvature strain will be introduced in the medium. The theory of such a deformation will be discussed at length in 3.2 for the present it will suffice to state some of the important results. The free energy per unit volume of the deformed medium relative to the state of uniform orientation is 

At the molecular level, it is obvious that curvature elasticity is a consequence of the orientational order in the liquid crystal. A quantitative relationship between them was established by Saupe using the mean field theory. From (2.3.6) is it seen that the dipole-dipole part of the dispersion energy of interaction of a molecule i in the average field due to its neighbours k is given by 

The internal energy per mole due to orientational order in the mean field approximation is then 

Since we are dealing with small deformations, we assume that the director continues to have cylindrical symmetry about the Z axis at the site of the th molecule and that the order parameter 5 is unchanged in magnitude. Substituting for etc. in (2.3.33) in terms of x defining the Eulerian angles between the molecular coordinate system and the X Y Z  

Now for a pure bend distortion Oa/OZ = 6a/0F = 0 and we may set sin aj. i jt0a/0Z. Assuming a spherically symmetric distribution, Xf /Rf etc. may be taken to be constant (independent of temperature), and therefore the second sum in (2.3.34) is proportional to The free energy of deformation per unit volume can then be expressed as 

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