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

12 CHAPTER 2. HAIRPIN CHAIN ELASTOMERSmatic director, n is Gaussian distributed and has a mean square displacementgiven by (§2.B)〈 〉R2 Lk B T⊥ = . (2.3)JNote that the mean square extent does not depend explicitly on the bendconstant. This is because although an increase in the bend constant results inan increase in the persistence length of each unit, this dependence is exactlycancelled out by the fact that the segments will have a smaller average tiltangle to the nematic field, and hence a smaller transverse component, as aresult of the increase cost of bending the polymer.2.2.3 Partition function for 1-D modelParallel to the nematic field, the end-to-end distance of the hairpin chain isprincipally governed by hairpins. Theend-to-end distribution can befound bycalculating the partition function of the hairpin chain. First imagine puttingn hairpin defects onto the polymer chain as shown in Fig. 2.5, using the followingprocedure: Begin creating a polymer chain in a highly ordered nematicenvironment, starting with the chain pointing in the up direction. Lay downthe polymer for a distance s 1 and then insert the first hairpin by changing tothe down direction. Then lay down a distance s 2 −s 1 and put in the secondhairpin by changing direction. This process is continued until all n hairpinsare been put in. Then put in the remaining piece of polymer so that the totalarc length is L. Repeat the process for all possible positions of the defectsalong the chain provided: s 1 < s 2 < s 3 < ... < s n < L.All defects are identical and so do pass through each other (to preventover-counting). This procedure can be expressed in an integral which givesthe number of configurations with a fixed end-to-end separation of R. Eachconfiguration that has the required end-to-end separation is counted as 1 byusing the delta function to impose the constraintR = s 1 −(s 2 −s 1 )...+(−1) n (L−s n ). (2.4)Integrating this constraint over all hairpin positions, and remembering thatthe chain can start in either the up or the down directions results in thefollowingΩ (n)± =∫ L0ds nl∫ sn0ds n−1l∫ s2ds 1...0 l)1× δ(2l [s 1 −(s 2 −s 1 )...+(−1) n (L−s n )∓R] , (2.5)where Ω (n)± denotes the number of configurations starting from the up (+) anddown (−) directions, L is the total arc length of the polymer and l is some