SIZE AND SCALING IN IDEAL POLYMER NETWORKS - EXACT RESULTS

The scattering function and radius of gyration of an ideal polymer net work are calculated depending on the strength of the bonds that form t he crosslinks. Our calculations are based on an exact theorem for the characteristic function of a polydisperse phantom network that allows for treating the crosslinks between pairs of randomly selected monomer s as quenched variables without resorting to replica methods. From thi s new approach it is found that the scattering function of an ideal ne twork obeys a master curve which depends on one single parameter x = ( ak)N-2/M, where ak is the product of the persistence length times the scattering wavevector, N the total number of monomers and M the crossl inks in the system. By varying the crosslinking potential from infinit y (hard delta-constraints) to zero (free chain), we have also studied the crossover of the radius of gyration from the collapsed regime wher e R(g) similar or equal to O(1) to the extended regime R(g) similar or equal to O(root N). In the crossover regime the network size R(g) is found to be proportional to (N/M)(1/4). The latter result can be under stood in terms of a simple Flory argument.