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

20 CHAPTER 2. HAIRPIN CHAIN ELASTOMERS1All Hairpins150.89Z hp0.60.470.253 Hairpins0-1 -0.5 0 0.5 1z/NFigure 2.7: A comparison between the partition functionsummed to infinity and just the first few terms. This is ata temperature satisfying fN = 8. The distribution is truncatedat z = ±N, the stretched out length of the chain.where the latter expression emphasises that the chains become rod like 〈z 2 〉 ∼N 2 unless the number of hairpins, fN, is large. At high temperature fN ∼ Nand on average there is a hairpin on every persistence length. In this temperatureregion the end-to-end span shows Gaussian scaling. At low temperatureswhere there are very few hairpins per chain fN ∼ 1 and the scaling is rod like.Fitting a Gaussian to the partition function provides a better estimate thanworking with the asymptotic approximation. The amplitude of the Gaussianis easily fixed by forcing agreement when z = 0Z GA (z) = Z hp (0)e −αN2 ( z N )2 , (2.46)where Z hp (0) = 2fI 0 (fN) + 2fI 1 (fN). To fit the curvature we need thefollowing property of modified Bessel functionsdI n (x)dx= 1 2 (I n−1(x)+I n+1 (x)). (2.47)For I 0 (x) we recall that: I n (x) = I −n (x). Fitting the curvature gives α =− Z′′ (0)2Z(0). The following expression for α is obtained(N 2 1 (fN)2α =I 0 (fN)(fN)I 0 (fN)+(fN)I 1 (fN) 4( (fN)2+ − fN ))I 1 (fN)+ (fN)2 I 2 (fN) . (2.48)2 2 4