Second and Fourth Moments of Chain Molecules

Abstract

Exact methods are developed for calculating second and fourth moments 〈M2〉 and 〈M4〉 of chain molecules, where M= ∑ i=1nmiand mi is a vector associated with Bond i but not necessarily colinear with it. The rotational isomeric-state approximation is adopted to represent the effects of hindrance potentials affecting bond rotations. No other approximations are invoked in deriving the statistical-mechanical averages denoted by 〈 〉. Account is taken of neighbor correlations, and the treatment is applicable alike to finite and infinite chains. Illustrative calculations of 〈r2〉/nl2 and of 〈r4〉/n2l4, where r is the end-to-end length of the chain, l the bond length, and n the number of bonds, are presented for normal alkane homologs H–(CH2)n–H, throughout the range of n from 1 < n < ∞. The methods are applicable also to chains having any variety of bonds in linear succession.