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

64 CHAPTER 3. POLARISATION OF CHIRAL ELASTOMERSvector quantities at the first stage of cross-linking. The fact that a n and r n0deform affinely can be used to express z as followsz = λ·λ −1f zf −tλ·λ −1f cf +tc. (3.37)This expression can then be substituted into the expression for the free energyand the variables r n0 , z f , and c f quench-averaged over, whilst annealing overc. The required averages of the distribution are denoted as follows〈z f z f 〉 = 1 3 j(n−j)a2 n l f (3.38)〈r o n0 ro n0 〉 = 1 3 na2 l 0 (3.39)〈c f c f 〉 = 1 3 Cf . (3.40)These averages are used to evaluate the expression∫ ∫ ∫ [∫f elk B T = − dc f P(c f ) dr n0 P(r n0 ) dz f P(z f )ln]dce − Fk B T. (3.41)Since the rigid-rod cross-link is short compared to the span of a Gaussianchain then the small elastic contribution of the rigid rod to the free energy isneglected. Terms linear in c f average to zero, terms in c and c f are small, andterms in c and z f are expanded down from the exponential and then averagedover w.r.t c and the resulting logarithmic term expanded.After evaluating these integrals the second stage of the cross-linking occursat a random point along the polymer chain, and consequently the randomlychosen j is averaged over. The expression for the free energy density obtainedis given byf el = 1 ( [2 n sk B T Tr λ·A·λ T ·l −1] −Tr[λ·B ·λ T ·l ∗−1])l ∗ = l·C −1 ·lA = (1−α)l 0 +αλ −1f·l f ·λ −Tf+αdλ −1f·C f ·λ −TfB = αdλ −1f·l f ·λ −TfHere α = n rigid /(n 1 + n rigid ) where n rigid is the number density of secondstage cross-links and n 1 is the number density of first stage cross-links. Thusα is the fraction of second stage cross-links. The constant d arises from averagingover the position j that the rod linker is bonded, and is given byd = [ 2ln(n−1)t 2] / [ (n−1)a 2] . After this second stage of cross-linking, therubber then relaxes to the optimal deformation. The total relaxation of therubber from its first cross-linking state to its current relaxed state is denotedby λ r . This relaxation is chosen so as to minimise the free energy density.