Expansion of a polymer chain with excluded volume interaction

Abstract

We present explicit values of the mean square end-to-end distance 〈R2〉 of a linear flexible chain for all values of the excluded volume parameter z>0 in space dimension 3. Our results are obtained by performing series analyses of our earlier sixth-order perturbation calculation using Borel summation technique and a consideration of the second virial coefficient. The philosophy of our series analyses is essentially the same as that of Le Guillou and Zinn-Justin [Phys. Rev. B 21, 3976 (1980)]. Changes in 〈R2〉 values with perturbation theory order and with technical variations in the analysis allow us to estimate the accuracy of our results. We make comparisons with many of the previously known crossover formulas. The results of des Cloizeaux et al. [J. Phys. Lett. (Paris) 46, L595 (1985)], obtained by an entirely different analysis of the same sixth-order series, is in excellent agreement with that presented here and well within our estimate uncertainty. Our results can be summarized throughout the full range of z by the simple formula ln(〈R2〉/Ll)=0.1772 ln(1+7.524z+11.06z2), where z=(3/2πl)3/2 wL1/2 with L the chain length, l the Kuhn length, and wl2 the effective binary cluster integral for a pair of segments.