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

62 CHAPTER 3. POLARISATION OF CHIRAL ELASTOMERSwhereβ =χ =αr(3.32)(1−r) 1r −1 n·λ·(l 0 +βδ (tr) )·λ T ·n. (3.33)Eq. (3.31) can then be substituted back in to the expression for the polarisation.The result of this yieldsP i = 1 ( ) 12 n sd(b/a)r −1 ǫ ijk (λ·(l 0 +βδ (tr) )·λ T ·nn) jk (3.34)This is zero because n is a principal axis of the tensor λ·(l 0 +βδ (tr) )·λ T , soit is a contraction of an antisymmetric tensor with a symmetric tensor.3.3.4 Rod cross-linkers introducing semi-softnessTypically, elastomers are aligned by a two-stage cross-linking process resultingin a monodomain. A model of this multistage cross-linking process [49] showsthat Gaussian chains do not remember the details of the strains imposed duringthe cross-linking process, indeed a multistage cross-linked network shouldexhibit the same behaviour as a single-stage system. This extended the ideasof Scanlan [50] into nematic elastomers. The process of multistage crosslinkingmay allow some extra orientational information to be imprinted intothe elastomer. A model of this process presented in [51] is now summarised,where the second stage is carried out using rod cross-linkers. The polarisationof the rubber within this model with this second stage of cross-linking willsubsequently be calculated.Two-stage cross-linkingConsider two separate strands of n monomers that have already been crosslinkedinto a matrix. These strands are then cross-linked together as shown inFig. 3.5. Initially the polymer chain was cross-linked with its ends at r 0 andr n . The rod cross-linker is joined to this original chain at the jth monomerdown its length. If we denote the position of the jth monomer by r j , and thedistance from the end to the jth monomer as: r j0 = r j − r 0 , then the freeenergy of the strand with its new cross-linking point can be written as thesum of the two Gaussian strands⎛⎞fk B T = 3 ⎝ rT j0 ·l−1 ·r j0 r T nj ·l−1 ·r nj2a 2 + ⎠. (3.35)j n−j