Crossover between Gaussian and self-avoiding limits via finite order self-avoiding walk: Conformation space renormalization for polymers. VII

The crossover between the random walk and the self-avoiding walk via finite order self-avoiding random walk (FSAW) is studied analytically with the aid of conformation space renormalization group theory. Explicit expressions for the end-vector distribution function and 〈R²〉, are given up to order ε=4-d (d the spatial dimension). Since the excluded volume parameter for FSAW is contour length dependent, it is very awkward to study the present crossover behavior by using the polymer-magnet analogy. In contrast, our calculations in conformation space are simple and transparent, showing the power of this RG approach. The crossover behavior along FSAW is compared with the crossover obtained when the magnitude of the excluded volume interaction is decreased. The crossover via FSAW may occur in situations where a very long single chain is immersed in a solution of (shorter) chains and the concentration of the (shorter) chains is increased.