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

104 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERS4.3.3 B < ∞The same uniform deformations as §4.3.1 are now considered but this timeallowing for a finite layer modulus. The finite layer modulus results in athreshold behaviour as when the elastomer is stretched parallel to the layernormal, because a cross over from stretching the layer spacing to rotation ofthe layers. In this section the microstructure associated with layer rotationwill not be considered. However, its effect on the elastic properties of thematerial are small because of the low energy cost of the interfaces betweendomains. The rotation of the layers combined with the microstructure (similarto that observed in nematic elastomers) provides an explanation of theHelfrich-Hurault effect observed in smectic LSCEs. Note that this is a differentphysical reason for the Helfrich-Hurault effect than in liquid smectics.In liquid smectics the undulations in the layers occur because of a competitionbetween bending the layers, penalised by the Frank elastic constants, andkeeping their spacing fixed. In liquid crystal elastomers the threshold is controlledby the competition between the energy cost of stretching the layers andshearing the elastomer – a rubber elastic effect. In the following subsectionsthe ratio of the layer modulus to the rubber modulus is denoted as: b = B µ .Imposed λ zzFor the case of finite layer modulus, it is assumed that the deformation matrixhas the same form as for the infinite layer modulus case⎛ ⎞λ xx 0 0λ = ⎝ 0 λ yy 0 ⎠. (4.104)λ zx 0 λFor a finite layer modulus the constraints are that of volume conservationdet(λ) = 1 = λ xx λ yy λ, (4.105)and of no rotation of the layers with respect to the matrixq = λ −T ·q 0 . (4.106)More care is required than in §4.3.1 when calculating the director as can nolonger simply write down the components from the cofactors. The directormust be normalised explicitlyn = (−√ λzx ,0, √λ xx(4.107)λ 2 xx +λ 2 zx λ 2 xx +λzx). 2The free energy density for the system can now be written down, includingthe new contribution from the finite layer modulus using the expression from