Segmental orientation in uniaxially deformed networks. A higher order approximation for finite chains and large deformations

The orientation of a reference vector m rigidly embedded in a chain of a deformed network is considered. As a first step, following Nagai's earlier treatment, the mean-square cosine 〈cos² θ〉_r of the angle that m makes with a laboratory-fixed axis is formulated for a chain with fixed end-to-end vector r. An expression including terms up to fifth inverse power of n, where n is the number of bonds in the network chain, is obtained for 〈cos² θ〉_r. Next, the corresponding average over all chains of a network and the associated orientation function, S, are found in terms of (i) unperturbed chain moments readily obtainable by the rotational isomeric state scheme and (ii) the extension ration of deformed networks. Such a rigorous expression for S is particularly useful for relatively short chains and for moderate to large deformations that cannot be satisfactorily accounted for by existing simpler formulations. Thus, we estimate ranges of extension ratios λ to which the conventional first-order approximation may be confidently applied. Calculations performed for polyethylene chains with n = 20-100 gave results that could be compared with those obtained in Monte Carlo simulations and previous theoretical approaches. These comparisons demonstrate the importance of adoption of a higher order approximation for the orientation function for λ ≥ 1.8.