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

2.2. MODEL OF HAIRPIN CHAINS 17The number of hairpins on the chain is governed by the temperature of thesystem and the energy of a hairpin defect. The partition function can becalculated by summing all the configurations multiplied by their associatedBoltzmann factors. This is most naturally done by splitting up the sums foreven and odd numbers of defects.Z hp = Z even +Z odd (2.26)Z even = ∑ 2N n−1 (n(n−2)!! 2 1− ( )n−2)z 2 2Ne −nβu h(2.27)even nZ odd = ∑odd n2N n−1 ((n−1)!! 2 1− ( zN) 2)n−12e −nβu h. (2.28)Fortunately these series can be summed up to infinity resulting in modifiedBessel functions. The power series expansion of modified Bessel function isgiven by∞∑I ν (z) = e −π 2 νi (z/2) ν+2kJ ν (iz) =(2.29)(k!)(ν +k)!k=0In summing the odd series, the following result is useful(n−1)!! = 2.4....(n−1) (2.30)( )= 2 n−1 n−12 ! (2.31)2Substituting n = 2p+1 reduces the odd part of the partition function to(∞∑ 1− ( ) )z 2 pN2pNZ odd = 2(p!) 2 2 2p e −(2p+1)βu h(2.32)p=0= 2e −βu hI 0(e −βu hNSimilarly for the even series, the result(n−2)!! = 2 n−22√1− ( ))z 2N. (2.33)( n−22)! (2.34)enables the series to be summed. Substituting n = 2p gives( ( ∞∑ N 2p−1 1− z) 2) p−1NZ even = 2e −2pβu h. (2.35)2(p−1)!p! 4p=1Substituting p ′ = p−1 enables this expression to be rewritten as( ( ∞∑ N 2p′ +1 1− z) 2) p ′NZ even = 22(p ′ +1)!p ′ e −2(p′ +1)βu h(2.36)! 4p ′ =0