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

72 CHAPTER 3. POLARISATION OF CHIRAL ELASTOMERSis⎛ ⎞λ xx 0 δλ = ⎝ 10λ zzλ xx0 ⎠ (3.59)0 0 λ zzwhere λ xz is denoted simply by δ and volume constraint has been included.The λ xz component has been suppressed on geometric grounds, since a λ zxdeformation is beingapplied byplates attached tothesample. Thefreeenergydensity of the mixture of chains is given by]f ss = 1 2[λ·l µTr 0 ·λ T ·l −1 +αδ (tr) ·λ T ·nn·λ , (3.60)where as usual α = 〈 1 r 〉 − 1〈r〉 and the l 0 and l −1 are in terms of 〈r〉. Theaverage value for the binary mixture of chains considered here is given by:〈r〉 = qr 1 +(1−q)r 2 , where q is the fraction of main chains of anisotropy r 1and 1−q is the fraction of side chains of anisotropy r 2 . On substituting intothis formula the director orientations n 0 = (0,0,1) and n = (sinθ,0,cosθ),then the following expression for the free energy density is obtained2f ssµ=1λ 2 xx λ2 zz+λ 2 zz(1+(r −1)sin 2 θ)+(rδ 2 +λ 2 xx)(1+( 1r −1 )sin 2 θ− (r −1)λ 2 zzδsin2θ +αλ 2 xxsin 2 θ (3.61)This free energy density is very similar to the case where the driving deformationis a stretch along the x axis and a shear is induced as a result of therotation of the director. As a result it is possible to re-express some of theresults for the imposed stretch case to apply here. For the case where there isno semi-softness the following solution is obtained{λ 2 xx = 1 2(r +1−δ 2 r)− √ }(r +1−δ 2 r) 2 −4r (3.62)λ yy = 1 (3.63)λ zz = 1/λ xx (3.64)sin 2 θ =rr −1λ 2 xx −1λ 2 . (3.65)xxThis is the solution up to the end of softness which occurs at δ cr = 1 − 1 √ r.After this point, minimisation of the free energy density becomes very difficultanalytically. The solution also becomes rapidly intractable when we have afinite value of α. To calculate the size of the polarisation in these cases anumerical procedure was used to minimise the free energy density. A simplexalgorithm provides a convenient and robust method for minimisation of thefree energy density. Fig. 3.10 shows a plot of the relaxation of λ xx , λ yy andλ zz for an elastomer consisting of only one type of chain, and for an elastomermade of a mixture of two sorts of chains. Note that the pureelastomers, α = 0)