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

60 CHAPTER 3. POLARISATION OF CHIRAL ELASTOMERSThe first of these is recognisable from the definition of l 0 . Using the factthat the first order shift in the principal axis is orthogonal to n 0 produces thefollowingr (1) = n 0 ·l ′ ·n 0 (3.25)δn (1) =l ′ ·n 0 −(n 0 ·l ′ ·n 0 )n 0(r −1). (3.26)When this first order shift is substituted into the polarisation expression thefollowing result is obtained[(P i ≈ ǫ ijk l 0 +l ′)·l −1] jk= ǫ ijk[(δ +(r −1)n 0 n 0 +l ′)·(δ +(1≈ ( ]1r −1) ǫ ijk[l ′ ·n 0 n 0 +(r −1)n 0 δnr+r (1) −1= ( 1r −1) ǫ ijk[l ′ ·n 0 n 0 +n 0(l ′ ·n 0 −(n 0 ·l ′ ·n 0 )n 0)]jk))](n 0 +δn)(n 0 +δn)jkjk(3.27)Where use has been made of the fact that l ′ is symmetric. The last term herevanishes by symmetry and the remaining two terms cancel out on interchangeof the suffixes j and k due to the total antisymmetry of the Levi-Civita symbol.Thus it is clear that the rotation of the director means that there is nopolarisation in equilibrium for a pure (i.e. not semi-soft) chiral elastomer.3.3.2 Vector properties from quadrupolesA more general argument as to why there is no polarisation can also be constructed.Consider two quadrupolar objects represented by the vectors k andn. A pseudo-vector can be made out of these two quadrupolar objects asfollowsp = α(n·k)(n×k). (3.28)Note that even powers of the two vectors are required by their quadrupolarsymmetry. It is clear from this object that if a pseudo-vector can be definedfrom the quadrupolar objects then the quadrupolar objects must not be eitherparallel or orthogonal to each other.Thespecificcase ofapiezoelectric responseofaliquidcrystallineelastomeris now considered. In this case the two quadrupolar objects are: the liquidcrystal order, as specified by the director n and a quadrupolar object definedby the elastic deformation we apply, M. To develop a polarisation the axesof these two objects must not be orthogonal. On minimising the free energydensity it is found that the director must sit along a principal axis of the