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

46 CHAPTER 2. HAIRPIN CHAIN ELASTOMERS2.E Spring constant of an extended worm-likechainThe spring constant of an extended worm can be calculated as follows. Thepartition function of a worm chain in a nematic field can be written as∫ {Z(f) = Du(s)exp −β 1 ∫ [L] }ds B∂u22 ∣∂s∣−Ju 2 z +βf ·R ,0where f is the applied tension. In a strong nematic field it is assumed thatthe direction of the polymer segments fluctuate around the director direction(ẑ). A small angle approximation can then be used by definingu = ẑ+σ +O(σ 2 ),where σ is a vector perpendicularto ẑ, required to keep u a unit vector. Usingthis substitution and working to first order in σ the partition function can becalculated as∫ {Z = Dσ(s)exp −β 1 ∫ [L( ) ]}∂σ2ds B +Jσ 2 +f z σ 2 ,2 ∂s0Where the delta function constraint has now been satisfied. Now a discreteFourier transform is performed on σ usingσ(s) = ∑ qσ(q)e iqswhere q is a scalar. The partition function is then given by∫ {}Z = Dσ(q)exp −β 1 ∑[ Bq 2 ]+J +f z |σ(q)|2.2qFrom the principle of equipartition of energy we have that each of the modeshas energy k BT2 . Consequently|σ(q)| 2 =k B TBq 2 +J +f z.By integrating over all q values the average value of |σ| 2 can be calculated〈|σ| 2 〉 = k B T∫ ∞−∞dq 12π Bq 2 =+J +f zk B T2 √ B(J +f z )Here it is assumedthat thesummation can beconverted to anintegral becausethe q values are so closely spaced. The limits of the integrand are taken to be