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

18 CHAPTER 2. HAIRPIN CHAIN ELASTOMERS= 2= 2√4∑∞1− ( )z 2N p ′ =0e −βu h√1− ( zNN 2p′ +1 e −2(p′ +1)βu h2(p ′ +1)!p ′ !) 2I 1(e −βu hN( (1−z) 2) p ′ +1/2N(2.37)4√1− ( ))z 2N. (2.38)Combining both of these results together with the delta functions that representthe chain in its fully stretched out configuration gives the full partitionfor the 1-D hairpin chainsZ hp (z) = 2 [ √N (fN) I 0(fN 1− ( ))z 2N( √1+ √1− ( )I 1 fN 1− ( ) ⎤ )z 2⎦z 2NN+ δ(z −N)+δ(z +N) (2.39)where f is the Boltzmann factor of a single hairpin f = e −βu h, and the partitionfunction has been expressed in terms of the natural variable fN. Thepartition function is of the formZ hp (z) = 1 h(fN,z/N). (2.40)NThe combination fN can be interpreted as approximately the average numberofhairpinsonahairpinnedchainwithz = 0(§2.D). Alternatively specificationof f gives the ratio of u h to k B T as βu h = −lnf. The expression for Z hp (z →N) is given byZ hp (z → N) → 2f +f 2 N (2.41)Thisissimplythesumofoneandtwohairpincontributions. Theotherhairpinformulae depend on the end-to-end distance whereas the one and two hairpinresults have no dependence on the end-to-end distance. When approximatingthis partition function, it is useful to know the value of Z hp (0)Z hp (0) = 2fI 0 (fN)+2fI 1 (fN) (2.42)Fig. 2.6 compares the end-to-end distributions for different temperatures expressedin the fN parameter. A comparison between the partition functionsummed over all hairpins and the partition function summed over only a finitenumber of hairpins is shown in Fig. 2.7. It shows that the partition functionconverges very quickly with the number of hairpin defects included in the sumfor small fN. At larger fN many terms are required for any sort of convergence.To develop a better intuition for this partition function and calculatesome of its limits it is useful to develop an approximation.