Estimation of High-Polymer Excluded Volume from Numerical Studies on Short Chains

The mean‐square length ${\mathrm{〈R}}^2〉$ of a polymer molecule composed of $n$ segments is assumed to behave asymptotically as ${\mathrm{〈R}}_n^2{\mathrm{〉\hspace{0.17em}\sim \hspace{0.17em}An}}^{\text{1+ε}}$ for large $n$ , where $A$ is a constant and $\epsilon$ takes into account the excluded‐volume effect. This paper compares calculations of ${\mathrm{〈R}}_n^2〉$ for the model of a polymethylene chain without excluded volume studied by Flory and Jerigan with the model of a polymer as a self‐avoiding walk on a crystal lattice which has been studied by the method of exact enumeration developed by Domb. For the first model $\text{ε\hspace{0.17em}=\hspace{0.17em}0}$ , whereas for the second, which includes the excluded‐volume effect, $\mathrm{\epsilon \hspace{0.17em}=\hspace{0.17em}}\frac{1}{5}$ . It is shown using the extrapolation technique of Neville tables that the exact‐enumeration method correctly predicts $\text{ε\hspace{0.17em}=\hspace{0.17em}0}$ for the Flory–Jernigan model. However, the rate of convergence is very much slower than comparable calculations using the fcc lattice and taking account of excluded volume.