Rheology of gelling polymers in the Zimm model

In order to study rheological properties of gelling systems in dilutesolution, we investigate the viscosity and the normal stresses in the Zimmmodel for randomly crosslinked monomers. The distribution of cluster topologiesand sizes is assumed to be given either by Erd\H os-R\'enyi random graphs orthree-dimensional bond percolation. Within this model the critical behaviour ofthe viscosity and of the first normal stress coefficient is determined by thepower-law scaling of their averages over clusters of a given size $n$ with $n$.We investigate these Mark--Houwink like scaling relations numerically andconclude that the scaling exponents are independent of the hydrodynamicinteraction strength. The numerically determined exponents agree well withexperimental data for branched polymers. However, we show that this traditionalmodel of polymer physics is not able to yield a critical divergence at the gelpoint of the viscosity for a polydisperse dilute solution of gelation clusters.A generally accepted scaling relation for the Zimm exponent of the viscosity isthereby disproved.Comment: 9 pages, 2 figure