Persistence vectors and higher moment tensors of polyoxyalkanes

Statistical mechanical averages of vectors and tensors characterizing the spatial configurations of polyoxymethylene (POM) and polyoxyethylene (POE) chains, i.e., the chain vector r connecting the ends of the chain and the tensors formed from this vector, are evaluated in internal reference frames attached to the first two bonds of each chain as functions of its length. Convergences of the persistence vectors a ≡ 〈 r 〉 with chain length to their limits a _∞ are delineated. For POE, the orientation of a _∞ with respect to the X₁ axes (the direction of the first bond) depends markedly on the choice of the initial bond and hence also on the internal frame of reference. Cartesian tensors up to fourth rank formed from the displacement vector ρ = r ? a are evaluated for n = 4–1922 bonds for POM and n = 3–300 bonds for POE. The second moment tensor 〈ρρ^T〉 calculated for POM is nearly cylindrically symmetric about the axis perpendicular to the plane of the first two bonds. Corresponding symmetry is absent in POE. Correlation with the freely jointed chain is investigated for the components of the tensors of fourth rank formed from the reduced vector \documentclass{article}\pagestyle{empty}\begin{document} $\tilde \rho = \left\langle {\rho \rho ^{\rm T} } \right\rangle ^{ - 1/2} \rho $\end{document}. Use of the equivalent chain model for this purpose is validated for POE but not for POM, even at n = 2000 bonds. At the limit 1/n = 0, the parameter m, representing the number of bonds of the real chain equivalent to one of the model, is estimated to be ca. 31 for POM. For POE a value of m = 10.3 ± 1.3 holds for n > 20 for all components. Results are compared with those calculated previously for polymethylene, poly(dimethylsiloxane) and polypeptides. The fact that values of m derived by analysis of moments are larger than those obtained in the usual manner by fitting the model chain to 〈r²〉₀ and r_max for the real chain demonstrates the inadequacy of artificial models for the representation of the real chain.