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Dynamical Eigenmodes of Star and Tadpole Polymers

The dynamics of phantom bead-spring chains with the topology of a symmetric star with $f$ arms and tadpoles ($f=3$, a special case) is studied, in the overdamped limit. In the simplified case where the hydrodynamic radius of the central monomer is $f$ times as heavy as the other beads, we determine their dynamical eigenmodes exactly, along the lines of the Rouse modes for linear bead-spring chains. These eigenmodes allow full analytical calculations of virtually any dynamical quantity. As examples we determine the radius of gyration, the mean square displacement of a tagged monomer, and, for star polymers, the autocorrelation function of the vector that spans from the center of the star to a bead on one of the arms.