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

84 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSof the director from the layer displacement. The layer normal is parallel to∇φ where φ is defined in Eq. (4.3). The free energy cost of bending the layerscan be calculated from the Frank elastic term associated with splay, but withrenormalised elastic constants compared with a nematic because of the layerformation. Assuming that ∇u is small then the following results are obtainedn x = − ∂u∂xn y = − ∂u∂yf = 1 2 B ( ∂u∂z) 2+ 1 2 K 1(4.14)(4.15)( ∂ 2 u∂x 2 + ∂2 u∂y 2 ) 2. (4.16)Note that the associated length scale ξ = (K 1 /B) 1/2 is of the order of the layerthickness, and that there can be no twist deformation because n·∇×n = 0by the commutative property of the partial derivatives for sufficiently smoothfunctions. Typically, the Frank elastic constant, K 1 , is larger than that ofnematics because of the higher degree of order in smectics.The Helfrich-Hurault effect requires the calculation of the contribution ofthe magnetic field to the free energy. The susceptibility tensor χ is uniaxialwith its principal axis along the director n, so the magnetisation (induced bythe external magnetic field) is given by: M = χ·H = χ ⊥ H+(χ ‖ −χ ⊥ )(H·n)n.The part of the magnetic free energy density that depends on the director isthus given byf mag = − 1 2 µ 0χ a (n·H) 2 , (4.17)where χ a = χ ‖ −χ ⊥ is the anisotropy in the susceptibility of the rods.4.1.4 The Helfrich-Hurault effect in smectic liquid crystalsThe Helfrich-Hurault effect has been analysed in [61] as follows. Consider anSmA liquid crystal between two plates, with the layers parallel to the walls.Applying a magnetic field to the system (Fig. 4.2) in the plane of the layersgenerates a torque that acts to turn the layers. The clamping at the walls willprevent the rotation of layers occurring because as soon as the layer rotationstarts there is an infinite energy cost as the layers pile up at the walls, andtheir spacing there collapses.An alternative to this bulk rotation is local rotation of different parts oflayers in opposite directions, so that the layers lower their energy with respectto the magnetic field whilst avoiding the cost associated with reducing thelayer spacing, as well as any global layer movement. Above a certain thresholdmagnetic field, H, the layers start to rotate. This can represented in terms ofthe layer displacement variableu(x,z) = u 0 (z)coskx. (4.18)