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

96 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSFrom this expression it is clear that the layer normal should deform accordingto the transpose of the inverse matrix, i.e. the matrix of cofactors.The layer spacing for this affine model can also be calculated. Considertwo adjacent planes in the deformed system, the first of which contains thepoint p and the second contains the point q as illustrated in Fig. 4.8. Fromdx+ n 0qd 0d 0pxOOFigure 4.8: The figure shows the required vectors to calculatethe spacing between the layers as the elastomer is deformed.Fig. 4.8 it is clear thatp = λ·x (4.58)q = λ·x+d 0 λ·n 0 (4.59)The displacement between these points resolved along the layer normal givesthe spacing between the smectic layersd = (q−p)·n (4.60)= d 0 (λ·n 0 )·n. (4.61)Assuming that the normal is initially parallel to the z axis then the layerspacing can be written asdd 0= (λ·n 0 )·=1|ǫ ijk λ jx λ ky |λ −T ·n 0|ǫ ijk λ jx λ ky |(4.62)(4.63)whereEq.(4.54) hasbeenusedtosubstitutefornandthedenominator rewrittenby identifying k = ˆx and m = ŷ. The cross product expression that producesthe new layer normal can be thought of geometrically as calculating thedistance along the normal between two planes, or equivalently as calculatingthe area of the plane. These two statements are equivalent because the systemis volume conserving.