Non-Gaussian Character of Real Polymer Chains

Abstract

It is shown that the distribution function of the end-to-end vector of a polymer chain at its free state as well as the partition function of the polymer chain subjected to an external force, may be expressed in terms of even moments of the end-to-end distance of the chain at its free state. In looking over these functions it is seen that the quantities g2k and g2k' (k≥2) in the text, representing the non-Gaussian character of a polymer chain, are related in definite ways with even moments. As a main contribution of this paper, a mathematical method is presented for calculating the even moments just described and in particular, the fourth moment 〈R4〉, by taking into account the interdependent internal rotation. For the sake of simplicity, polyethylene is employed to illustrate the method. The expression for 〈R4〉 is obtained as a simple function of the (1, 1) elements of several matrices. The N (N is the number of skeletal bonds) dependence of g2k and g2k', and thereby also that of every term in a series expansion of the distribution function, are discussed in full detail.