The number of contacts in a self-avoiding walk of variable radius of gyration in two and three dimensions

A simple scaling law is proposed for the dependence of the number of contacts on the radius of gyration of a self-avoiding walk (SAW). We test our proposal on SAWs generated by a Monte Carlo simulation on square and cubic lattices. The distribution of the number of contacts is then combined with the distribution of configurations previously derived to deduce the free energy of a polymer chain for low values of the interaction parameter χ. As compared to the free energy of Flory, the new expression takes a much better account of the spatial correlations between distant monomers of the chain.