  We investigate the effect of the number of arms (functionality f ) on the
mobility of the branch point in symmetric star polymers. For this purpose we
carry out large-scale molecular dynamics simulations of simple bead-spring
stars and neutron spin echo (NSE) spectroscopy experiments on center labeled
polyethylene stars. This labeling scheme unique to neutron scattering allows us
to directly observe the branch point motion on the molecular scale by measuring
the dynamic structure factor. We investigate the cases of different
functionalities f = 3, 4 and 5 for different arm lengths. The analysis of the
branch point fluctuations reveals a stronger localization with increasing
functionality, following 2/f-scaling. The dynamic structure factors of the
branch point are analyzed in terms of a modified version, incorporating dynamic
tube dilution (DTD), of the Vilgis-Boue model for cross-linked networks [J.
Polym. Sci. B 1988, 26, 2291-2302]. In DTD the tube parameters are renormalized
with the tube survival probability \phi(t). As directly measured by the
simulations, \phi(t) is independent of f and therefore the theory predicts no
f-dependence of the relaxation of the branch point. The theory provides a good
description of the NSE data and simulations for intermediate times. However,
the simulations, which have access to much longer time scales, reveal the
breakdown of the DTD prediction since increasing the functionality actually
leads to a slower relaxation of the branch point.
