ONE-PARAMETER RECURSION MODEL FOR FLEXIBLE-CHAIN POLYMERS

A simple set of recursion relations determines the radius of gyration and second virial coefficient of flexible-chain polymers of polymerization index 2N from those at N. In the limit of small binary excluded volume, these recursion relations accurately reproduce the known two-parameter model results, and for large binary excluded volume such as occurs in lattice self-avoiding walk models, the agreement remains good to a few percent. "Direct renormalization" differential equations similar to those introduced by des Cloizeaux (J. Phys. (Paris) 1981, 42, 635) are found to be very nearly equivalent to the recursion relations and have the advantage that an explicit solution is possible. Both the recursion relations and differential equations clearly illustrate the limitations of the two-parameter model and why the experimentally observed interpenetration function is often seen to decrease with increasing polymerization. Perhaps more surprisingly, they serve to show that it is incorrect to identify the increase in the interpenetration as a function of temperature at fixed polymerization with that observed in the two-parameter model.