Distribution Functions for Chain Molecules

Abstract

Distribution functions $\mathrm{W(}\mathrm{r})$ of the end‐to‐end vectors r of chain molecules are calculated according to: (i) the exact expression of Rayleigh for a freely jointed chain, (ii) the Kuhn and Grün approximate function, (iii) a revised form of the KG function which amends a basic error in its traditional derivation, and (iv) the Gaussian function $W_{\infty }(\mathrm{r})$ . For chains of 4–20 freely jointed bonds, (iii) is a considerable improvement over (ii). Except at higher extensions, in the vicinity of $\text{r\hspace{0.17em}/\hspace{0.17em}nl≈1}$ , the Gaussian expression (iv) offers the most satisfactory approximation to (i). Series expansion of $\mathrm{W(}\mathrm{r}{\mathrm{)\hspace{0.17em}/\hspace{0.17em}W}}_{\infty }(\mathrm{r})$ in Hermite polynomials in the even moments ${\mathrm{〈r}}^2{〉}_0$ , ${\mathrm{〈r}}^4{〉}_0$ , etc., readily yields results previously obtained by a more lengthy Fourier transformation method due to Nagai. For chains of short or moderate length the convergence of Nagai's series is slow, many higher moments being required for significant refinement of $\mathrm{W(}\mathrm{r})$ beyond its approximation by $W_{\infty }(\mathrm{r})$ . Analysis of the moment ratios ${\mathrm{〈r}}^4{〉}_0{\mathrm{\hspace{0.17em}/\hspace{0.17em}〈r}}^2{〉}_0^2$ and ${\mathrm{〈r}}^6{〉}_0{\mathrm{\hspace{0.17em}/\hspace{0.17em}〈r}}^2{〉}_0^3$ and their trends with chain length suggests that the Rayleigh (exact) distribution functions for freely jointed chains may afford very good approximations for real chains, provided, however, that the number of equivalent bonds for the freely jointed model is properly scaled in relation to the actual number of bonds in the real chain. About 20 bonds of a polymethylene chain are equivalent to one of the freely jointed model chain.