Angular correlation ranges in flexible polymer chains

Abstract

We consider the equilibrium structure of flexible polymer chains within the rotational isomeric state model. This model is solved by an irreducible tensor method which is somewhat different from, and simpler than, the approach of Flory and others. We evaluate in particular the quantities 〈DJmm′(Ωn) 〉, where DJmm′ is the mm′th component of the Jth rank Wigner rotation matrix. Ωn is the orientation of the nth monomer with respect to the first, and the brackets denote a canonical ensemble average over all polymer conformations. We find—for physically reasonable values of the parameters defining the three-state rotational isomeric state model—that 〈DJ=1,2oo(Ωn) 〉 decays to zero as (0.632) (±0.841)n−1 for large n. 〈DJ=3,4oo(Ωn) 〉, however, is anomalously long ranged. When the complement of the valence bond angle is tetrahedral (cosϑ=1/3), in fact, 〈DJ=3oo(Ωn) 〉∼ (0.555) (−1)n−1 for large n. This conspicuously unphysical behavior is shown to be an artifact of the three-state model for rotational isomers. Finally, we use our results for averages of the form 〈D2oo(Ωn) 〉, 〈r2nD2oo (Ωn) 〉 and 〈r2nD2o-m(Ωn) D2-mo(φ?nϑ?n0) 〉 to compute the light scattering intensities (I) from dilute solutions of flexible polymer chains. (Here rn is the vector between the centers of mass for the first and nth monomers.) We find negligible angle dependence for I, and (IVh/IHh)90°=1 to within one part in 105.