Statistical models of elasticity in main chain and smectic liquid ...

3.3. THE POLARISATION OF A PURE LIQUID CRYSTALELASTOMERS 65After this relaxation, a further deformation on the rubber can be imposed,which corresponds to the deformation applied experimentally after the secondstage cross-linking process. A prime ( ′ ) is now used to denote quantities suchas X ′ = λ r ·X ·λ T r . The imposed deformation after the relaxation is denotedby λ. The final formula for the free energy density is thusf el = 1 2 n sk B T( [Tr λ·A ′ ·λ T ·l −1] [−Tr λ·B ′ ·λ T ·l ∗−1]) (3.42)Polarisation of two-stage cross-linked materialThe polarisation of an elastomer composed of chiral main chains after a twostage cross-linking process is now calculated. The polymer chains have chiralmonomers that carry a dipole moment, but the rod cross-linkers carry nodipole moment. The average binormal vector can be calculated as shown in§3.1.4. The original strand is split into two strands after the rod cross-linkeris attached to the polymer chain. The end-to-end vector of these two strandsis denoted by g o and g n as shown in Fig. 3.6. The total total binormal vectorg0gnFigure 3.6: The two end-to-end vectors of the polymer chainafter it has been cross-linked to the rod.for each strand can then be calculated using〈V〉 = 3bN 〈 〉2aL 2 g×l −1 ×g(3.43)This average can be calculated by first substituting for g in terms of z andr n0g 0 + j n r n0 +z = 0(g n − 1− j )r n0 +z = 0n