On the relationship between the characteristic ratio of a finite chain, C_n, and the asymptotic limit, C_∞

Often experiments with chains containing a finite number of bonds, n, are interpreted with the assumption that the characteristic ratio, C_n, is determined completely by C_∞ and n. This assumption is supported by some, but not all, textbook models for simple flexible chains. The freely jointed chain, freely rotating chain with fixed bond angle, and simple wormlike chain predict C_n = f(C_∞n). These three models share the feature that the stiffness of the chain is specified by no more than one parameter. However, when more than one parameter affects the stiffness of the chain, as in the model with fixed bond angle and symmetric hindered rotation about independent bonds, C_n is no longer determined by C_∞ and n alone, C_n ≠ f(C_∞,n). Since virtually all real chains have hindered rotation, they cannot be expected to have the dimensions given by C_n = f(C_∞, n). This conclusion is supported by numerical calculations using previously published rotational isomeric state models for polyethylene, polyisobutylene, and poly(dimethylsiloxane). Although these three polymers have similar values of C_∞, they may have quite different values of C_n. This conclusion from the calculations is consistent with the observed behavior of polyisobutylene and poly(dimethylsiloxane), as reported by Arbe et al. (Macromolecules 2001, 34, 1281) and by Sluch et al. (Macromolecules 2003, 36, 2721). The finite n effect in these polymers is three times stronger for the mean square unperturbed radius of gyration than for the mean square unperturbed end-to-end distance.