Distortions of liquid crystals

The three kinds of deformations are associated with the variation of n, i.e. Vn. For the splay deformation, the divergence of the n vectors, V n, is not zero for the twist deformation, n V x n 0, and for the bend deformation, n x V x n 0. In order to describe the meaning of the three formulae, it is supposed that in the undeformed sample, n points along the z direction. These three deformations can hence be written in the form of components as follows 

The strain increases the energy of the solid as a stress is applied. The distortion of the director in liquid crystals causes an additional energy in a similar way. The energy is proportional to the square of the deformations and the correspondent coefficients are defined as the splay elastic constant, K, twisted elastic constant K22 and bend elastic constant Kx, i.e., the respective energies are the half of 

The elasticity theory of liquid crystals was proposed by Oseen (1933) and Zocher (1933), and then modified by Frank into the form that has since 

Because of the spontaneously helical structure in the cholesteric liquid crystals, the second term in the integral must be revised to 

The form of free energy for smectic liquid crystals is different. If there are no defects in the smectic liquid crystals, the curl of n, V x n, must be zero. Thus, no twist and bend deformations exist in the smectic liquid crystals. In addition, there is an energy penalty associated with the translational deformation. For example, the displacement of smectic layer u will cause an additional term of elastic energy 

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