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

94 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSThe same procedure as that used for the previous integral is used; first thesum over y is evaluated picking out the particular value y = 12π qT 0 · Q, andthen the integral over Q performed. After carrying out the sum over y theresult is∫ {1dQ exp − 3 }N 2L Q·l−1 0 ·Q[ 3·2L QT ·λ T ·l −1 ·λ·Q+ β ]2 (qT 0 ·Q−qT ·λ·Q) 2 (4.48)The integral over Q can then be performed. The first term results in theusual trace formula expression. The second term can be evaluated using theaverage: 〈Q T Q〉 = 1 3 Ll 0. The result is thenLβ[ ()]6 Tr l 0 · q 0 ·q T 0 +λ T ·q·q T ·λ−λ T ·q·q T 0 −q 0 ·q T ·λ(4.49)This expression can be simplified by using the following definition of l 0 givenin Eq. (4.38). Since n 0 and q 0 are parallel this simplifies to[ (2π ) ]Lβ 26 Tr ld ‖ δ +l 0 ·λ T ·q·q T ·λ−λ T ·q·n T 2π0 l0 d ‖ − 2π l0 d ‖ n 0 ·q T ·λ0This expression can be rearranged intoLβ6((l 1/20 ·λ T ·q)− 2πd 0n 0√l ‖) 2(4.50)It can be seen from this expression that this constraint penalises q if it is notequal to λ −T ·q 0 . The resulting terms from the Q integral are thus12{Tr[λ·l 0 ·λ T ·l −1] + Lβ ((l 1/20 ·λ T ·q)− 2π √ ) } 2n 0 l3 d ‖0(4.51)Thus our final microscopic model for the smectic liquid crystal elastomer is ofthe formf = µ 2 Tr [λ·l 0 ·λ T ·l −1] + 1 2 B ( dd 0−1) 2, (4.52)where µ = k B Tn s . The second term is the layer compression penalty from thesmectic free energy density. The identification of the layer normal, q with thedirector, n, is rigidly made here, that is in the notation of [72] b ⊥ → ∞. Theconstraint: q = λ −T ·q 0 is rigidly imposed.