Statistical mechanics of helical wormlike chains. VIII. Angular correlation functions

The adaptation of the helical wormlike chain to a given real chain, or its rotational isomeric state model on the monomer-unit level, is studied with angular correlation functions between two localized coordinate systems affixed to the chain or between two monomer units as functions of their separation along the chain, as defined as the expansion coefficients of the Green’s functions. Since these correlation functions for the latter model depend on the orientation of a localized system affixed to the monomer unit with respect to that unit, they are first formulated in the Flory system and then transformed to those in a different system appropriate for a comparison with the helical wormlike chain. This transformation is determined to preserve certain symmetry properties of the helical wormlike chain in the correlation functions for the rotational isomeric state model. Numerical computations are carried out for polymethylene, polyoxymethylene, poly(dimethylsiloxane), tactic polystyrene, and tactic poly(methylmethacrylate) chains. The analysis shows that the decay of the torsional correlation in the helical wormlike chain is slow, while the higher order correlation functions for the rotational isomeric state model decay slowly. The imaginary parts of the correlation functions are related to the asymmetry of the chain, and their symmetry properties for the two models do not completely agree with each other in the case of the asymmetric chains. However, the imaginary parts are of minor importance since their magnitudes are small compared to the real parts; in particular, they vanish for the symmetric chains. As a whole, there is rather good agreement between the two models. The analysis provides also a general procedure of determining the orientation of the localized system of the helical wormlike chain with respect to the monomer unit, and therefore also to, for instance, the principal axes of the polarizability tensor of the latter.