The end-to-end distribution function for a flexible chain with weak excluded-volume interactions

TL;DR

This paper derives an explicit distribution function for a flexible chain considering weak excluded-volume interactions, revealing how these interactions influence chain statistics beyond mean-field approximations.

Contribution

It provides a novel explicit Green function approach for weak intra-chain interactions, capturing the regular dependence on interaction strength and the singular dependence on cutoff length.

Findings

Distribution function depends regularly on interaction energy ratio

Mean square end-to-end distance increases linearly with interaction strength

Dependence on cutoff length is singular and does not affect the leading term

Abstract

An explicit expression is derived for the distribution function of end-to-end vectors and for the mean square end-to-end distance of a flexible chain with excluded-volume interactions. The Hamiltonian for a flexible chain with weak intra-chain interactions is determined by two small parameters: the ratio $\epsilon$ of the energy of interaction between segments (within a sphere whose radius coincides with the cut-off length for the potential) to the thermal energy, and the ratio $\delta$ of the cut-off length to the radius of gyration for a Gaussian chain. Unlike conventional approaches grounded on the mean-field evaluation of the end-to-end distance, the Green function is found explicitly (in the first approximation with respect to $\epsilon$). It is demonstrated that (i) the distribution function depends on $\epsilon$ in a regular way, while its dependence on $\delta$ is singular, and (ii) the leading term in the expression for the mean square end-to-end distance linearly grows with $\epsilon$ and remains independent of $\delta$.