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

76 CHAPTER 3. POLARISATION OF CHIRAL ELASTOMERS1.5q = 0.0q = 1.01.0θ0.50.00.0 0.5 1.0 1.5 2.0λFigure 3.13: The figure shows a plot of the angle of the directorto the z axis as a function of the applied shear, λ whenno relaxation is allowed. The different curves correspond todifferent mixtures of the main chain and side chain polymers.The fraction of main chain is denoted by q.be calculated and is given by the expression2P yn s d b aq(1−q)(r 1 −1)(r 2 −1)(r 1 −r 2 )〈r〉λ={ ()}1r 1 r 2 (〈r〉−1) 4〈r〉 2 λ 2 + 1− q(1−q)(r 1−r 2 ) 2 2r 1 r 2 (〈r〉−1)+〈r〉(λ 2 2−1)(3.70)where 〈r〉 = qr 1 +(1−q)r 2 . This expression tends to have a maximum aroundλ = 1. The expression also simplifies markedly for the composition suchthat 〈r〉 = 1. In this case the director does not move under the shear. Thepolarisation expression then simplifies to2P yn s d b a= (r 1 −1)(r 2 −1)λ(r 1 −r 2 )(3.71)The director now remains fixed along the z-axis in this case as the shear iscarried out. This is now very similar to the toy model of [48].3.5 ConclusionsAnalysis of the symmetry of the free energy density of a chiral liquid crystalelastomer shows that the elastomer can develop a polarisation. One way itcan do this is similar to the flexoelectric effect in liquid crystals. An alternativemechanism is via a main chain polymer composed of chiral monomers.