Rotational Isomeric theory of macro molecules

A polyethylene chain which consists of single C-C bonds, undergoes internal rotation in each link of the chain. It is precisely this conformational change that decides the chain flexibility which is responsible for the rubber elasticity of polymers. We construct a simplified model of the macro molecule in the form of freely jointed rods of the same length b and each rod rotates freely relative to the adjacent ones. Therefore a set of conformations that arise from rotations around a given atom of the freely jointed chain is continuous in the range of angles from 0 to 4π and the energy is not changed on rotations. The chain can be characterized by a vector h drawn from the fit atom in the chain to the last one. It is evident that the value of the vector h averaged over all the conformations is equal to zero i.e. h average=0. Since on thermal motions, all its directions are equally probable, one must find out in this case, the distribution of the probabilities of realization of various erical value of h which can vary from 0 to the maximum length of the extended chain, zb.The solution of this problem is analogous to the solution in the problem in the theory of Brownian motion namely the finding of the probability of a particle being displaced to a distance h as a result of z steps each of which has a length b and an arbitrary direction.

Similar problem when A drunkard walks randomly from a starting point. Each of his steps is of length b. What is the probability of his  being found out at a distance between h and h+dh from the starting point after he has taken z steps ? It may be said that the steps   taken by the drunkard are quite random and hence obey Gaussian distribution law. The problem is similar to the problem of Brownian motion.

If W(h) is the probability of the chain ends lying at a distance between h and h+dh from the start, then

W(h) dh =                             (3/2πzb2)3/2 4πh2exp(-3h2/2zb2)dh

Probability Density=P(h) =  (3/2πzb2)3/2 *      exp(-3h2/2zb2)

Mean Square end-to-end distance = √[(h2)mean] =0h2 P(h)dh = zb2

RMS end-to-end distance=(√z)b

RMS Distance covered by a drunkard after z steps is directly proportional to square root of no. of steps.

Distance at which probability is maximum is given by h2  = 2zb2 /3;

 no. of steps (z) Length of each step (b) Expectation of distance covered (h) RMS end-to-end distance : Ist term in Blue 2nd term in black 3rd term in red Probability of covering expected distance(remaining in between h to h+dh) Maximum probability at distance Probability Density