Asymptotic Solution of the Excluded-Volume Problem in a Linear Polymer Chain

An asymptotic form of the expansion factor $\alpha$ for the end‐to‐end distance of a linear polymer chain is derived, based on the self‐consistent‐field approach developed recently by Reiss. Reiss' integral equation for the distribution function is corrected, since some factor in the transition probability is dropped. The self‐consistent potential is evaluated in the perturbed state by introducing the uniform‐expansion approximation. In addition, the present potential is appropriate for use near the peak of the distribution function, whereas his potential is valid at extremely large distances. The corrected integral equation is equivalent to that of James except that the present potential is evaluated in the perturbed state. The present theory predicts that at large ${\mathrm{z,\; \alpha }}^5\text{\hspace{0.17em}=\hspace{0.17em}1.45z}$ , where $z$ is the excluded‐volume parameter. An approximation that leads to an equation for $\alpha$ of the fifth‐power type, as suggested first by Flory, is discussed in detail.