Moments and distribution functions for polymer chains of finite length. II. Polymethylene chains

Abstract

Statistical mechanical averages of vectors and tensors characterizing the configurations of polymethylene chains with n = 5–200 bonds have been calculated, these quantities being expressed in the internal reference frame attached to the first two bonds. Specifically, the persistence vector a ≡ 〈r〉 and the averaged center‐of‐mass vector 〈g〉 have been obtained as functions of chain length. They are compared to the corresponding vectors for various model chains. Configurational averages of the Cartesian tensors formed from ρ = r − a have been computed up to and including the tensor of seventh rank. Density distribution functionsWa (ρ), evaluated from these moment tensors using the three‐dimensional Hermite series elaborated in the preceding paper, are approximately cyclindrically symmetric about a principal axis that lies close to the direction of a. The second moment (and, hence, the dispersion) along this axis is smaller than the moments perpendicular thereto. Acentricity and skewness of the distribution Wa (ρ) is large for n < 40 bonds; they are appreciable even at n = 100 bonds. For n ≥ 50, moments of the normalized density distribution can be represented in good approximation by a model chain consisting of freely jointed segments originating at the terminus of a, with each segment taken to be equivalent to 20 bonds of the real chain. The traditional approximation of the unnormalized density distribution about r = 0 by a freely jointed equivalent chain is subject to much greater error. The second moment tensor of gyration, averaged over all configurations, is approximately cylindrically symmetric, but its component (and its dispersion) along the symmetry axis is greater than in the transverse directions. This tensor is contrasted with that obtained by summing moments along the principal axes for individual configurations.