Model of liquid crystal networks

Now we return to liquid crystal networks. In addition to the above three postulates, we further assume (4) The total free energy of the network is comprised of elastic, nematic and external stress contributions, i.e., 

The nematic networks are composed of strands whose conformation is uniaxial with two end-to-end distance components not all being equal which read 

Under the Gaussian assumption, the probability of the end-to-end distance vector being R is 

For uniaxial deformations, two distinct principal extension ratios Ax and Ap, along and perpendicular to the director respectively, satisfy the relation Xz Xp = 1 because of the incompressibility of rubbers. 

If the chains are crosslinked in the isotropic state and Pq is an isotropic Gaussian distribution, the additional elastic free energy of a strand of the network with deformed span R is given by (denoting Ax by A) 

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