# Regensburg 2002 – wissenschaftliches Programm

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# DY: Dynamik und Statistische Physik

## DY 52: Weiche Materie

### DY 52.5: Vortrag

### Freitag, 15. März 2002, 12:15–12:30, H2

**Effective diffusion constant in the Zimm model for polymeric liquids at the gelation transition** — •Matthias Küntzel, Peter Müller, and Annette Zippelius — Institut für Theoretische Physik, Georg-August-Universität, D-37073 Göttingen

Effects of the hydrodynamic interaction on the critical dynamics of
gelling polymeric liquids are analyzed by means of a
generalized Zimm model with randomly distributed harmonic crosslinks
which are drawn from a mean-field distribution.
The main focus lies on the disorder-averaged effective
diffusion constant
*D*_{eff} defined via the long-time decay of the
incoherent scattering function.
Thanks to the preaveraging approximation *D*_{eff}^{−1}
decomposes into contributions ⟨ *D*^{−1}⟩_{n}
from averages over all polymer clusters of fixed size *n*.
Given the asymptotic power-law growth ⟨ *D*^{−1}⟩_{n}∼
*n*^{β} with cluster size, it follows that *D*_{eff}
vanishes with exponent 2β−1 at the gelation transition provided that
β ≥ 1/2. Otherwise *D*_{eff} does not vanish.
Exact analytical arguments yield the lower bound β≥ 1/4.
Numerical data for β lie closely above this lower bound, but well
below 1/2, the Zimm value for linear chains. Thus, the results show
that the diffusion constant of a cluster of size *n* depends not only
on *n*—as in the Rouse model—but also on the topology of the
cluster. Consequently, common heuristic approaches to gelation in the Zimm
model based on a combination of cluster statistics and properties of
linear chains are bound to fail.