Persistence length and finite chain length effect on characteristic ratios

Characteristic ratios for an unperturbed flexible chain with n bonds of length l can be defined with the mean-square end-to-end distance or the mean-square radius of gyration, C _r,n ≡ 〈r ²〉 ₀/nl ² and C _s,n ≡ 〈s ²〉 ₀/nl ², respectively. Their approaches to the asymptotic limits at n → ∞are found to be related by [dC _r,n/d(1/n)] _1/n=0 = 2{[dC _s,n/d(1/n)] _1/n=0 - C _s,∞}. Real chains usually have C _r,∞ > 1 and [dC _r,n/d(1/n)] _1/n=0 < 0, signifying that C _r,n makes its final approach to C _r,∞ from below. However, some chains with C _r,∞ > 1 have values of C _r,n that pass through a maximum at finite n and approach C _r,∞ from above, [dC _r,n/d(1/n)] _1/n=0 > 0. Syndiotactic poly(methyl methacrylate) (sPMMA) is an example. In general, [dC _r,nd(1/n)] _1/n=0 = -2Σ∞ _k=1 ^〈ku _iu _i+k〉 ₀, where u _i = l _i/l _i denotes a unit vector parallel with the vector for bond i. Members of this sum with 15 < k < 34 produce [dC _r,n/d(1/n)] _1/n=0 > 0 in sPMMA. These results have strong implications for the extraction of a "persistence length" from experimental data in the form 〈s ²〉 ₀/n as a function of n.