Statistical mechanics of helical wormlike chains. XIV. Radius and shape of branched chains

Abstract

Analytical expressions for the mean-square radius of gyration chains, such that the initial orientations (Euler angles) of branches at branch points are fixed with respect to other branches (of the star) or to the main chain (of the comb), corresponding to real branched chains. It is shown that for regular stars, the factor g, as defined as the ratio of 〈S2〉 of the branched chain to that of the linear chain of the same total contour length L, is independent of the relative initial orientations of branches for the reduced contour length Lb of each branch greater than about 100. In particular, for regular stars of rather strong helical nature, the behavior of g is shown to be similar to that of 〈S2〉/L for the corresponding linear chains; it first increases to its maximum with increasing L and then decreases monotonically to its coil limiting value. This behavior does not correlate with the value of the characteristic ratio C∞ of the corresponding real linear chain. A comparison of the present model is made with the rotational isomeric state model with respect to g for regular stars, taking as examples polymethylene, polyoxymethylene, and poly-DL-alanine. The agreement (and the disagreement) between the two models is discussed rather in detail. The moments of inertia tensor (its mean principal values) are also evaluated for regular stars approximately by replacing its three invariants by their averages as in the case of linear chains. It is then shown that the molecular shape depends more strongly on the relative initial orientations of branches at small L than does g, and is independent of them for Lb≳10, and that at a given L in the latter range as well as in the coil limit, the spherically symmetric shape is attained gradually with increasing branching.