Crossover behaviour between Gaussian and self-avoiding limits of a single polymer chain: Conformational space renormalisation for polymers. VI

The Gell-Mann-Low style conformational space renormalisation method for polymers is generalised to describe the crossover between the random walk and self-avoiding walk limits, i.e. to describe the excluded volume dependence. Explicit calculations are provided to order epsilon =4-d (d the spatial dimensionality) for the full end-to-end vector distribution function, the coherent elastic scattering function, the second virial coefficients and (R ²) and (S²). The crossover functions are required therefore to exhibit the correct asymptotic limits of both the random and self-avoiding walks. The theory demonstrates that the latter choice implies that the expansion factors, alpha ² and alpha _s², for the mean square end-to-end vector (R²) and radius of gyration (S²), respectively, are not universal functions of the single scaling variable describing the strength of the excluded volume interactions. Nevertheless, much of the available experimental data on long chain polymers appears to involve small renormalised dimensionless excluded volume, and therefore alpha ² and alpha _s² are approximately universal quantities. Comparisons between our theoretical predictions and experimental data on the second virial coefficient and alpha _s² show good agreement.